Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
Jean Moulin Ollagnier
anvd Statistical h/lechanics
Spri nger-Verlag Berlin Heidelberg New York Tokyo
Author Jean Moulin Ollagnier Departement de Mathematiques, Universite Paris Nord Avenue J. B, Clement, 93430 Villetaneuse, France
AMS Subject Classification (1980): 20F, 28D, 54H20, 82A05, 82A25 ISBN 3-540-15192-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15192-3 Springer-Verlag New York Heidelberg Berlin Tokyo Th~s work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft WOW', Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
INTRODUCTION
....................................................
I. P R E L I M I N A R Y
ANALYSIS
I.I.
Sublinear
1.2.
Compact
1.3.
Radon measures
1.4.
Extremal
1.5.
References
2. D Y N A M I C A L
........................................
functions
convex
and
sets
the H a h n - B a n a c h
theorem
.......
...................................
........................................
points
in c o m p a c t
convex
sets
................
............................................
SYSTEMS
AND AMENABLE
GROUPS
.......................
I I 4 6 10 14
15
2.1.
Dynamical
systems
2.2.
The
point
2.3.
Amenability
2.4.
References
............................................
33
3. E R G O D I C T H E O R E M S
............................................
35
4.
fixed
.....................................
V
property
and the a m e a n i n g
and a l g e b r a i c
constructions
3.1.
Invariant
linear
functionals
3.2.
Invariant
vectors
and mean
3.3.
Individual
ergodic
theorems
3.4.
The
ergodic
theorem
3.5.
References
ENTROPY
saddle
Equivalence
DYNAMICAL
4.2.
Entropy
of p a r t i t i o n s
Entropy
of d y n a m i c a l
4.4.
The
a n d the 4.5.
theorems
...........
41 47
............................
50
ergodic
52
.......................
53
systems ..............
53
................................. systems
subadditive
35
...........................
dynamical
Shannon-McMillan
References
19 30
..........................
ergodic
SYSTEMS
of a b s t r a c t
4.3.
almost
......
...............
............................................
OF ABSTRACT
4.1.
filter
15
..........................
54 59
theorem
theorem
......................
............................................
64 70
IV
5. E N T R O P Y
6.
7.
A
FUNCTION
AND
THE
5.1.
Topological
entropy
5.2.
Pressure
a continuous
5.3.
Entropy
5.4.
References
STATISTICAL
of as
MECHANICS
6.2.
Cocycles
6.3.
Phase
6.4.
Supermodular
6.5.
References
ON
of
and
and
72
.................
82
measure
measures
measures
86
................
91 96
............................
...........................................
SYSTEMS
IN
STATISTICAL
MECHANICS
7.2.
Invariant
Gibbs
measures
7.3.
Mixing
7.4.
Example:
7.5.
References
properties
and
.................
....................... equilibrium
measures
of
Ruelle's
....
.......................
...........................................
COUNTABLE
8.1.
Tiling
amenable
8.2.
Equivalence
8.3.
Rokhlin's
8.4.
References
of
AMENABLE
groups countable
lemmaand
GROUPS
...................
............................... groups
......................
hyperfiniteness of countable amenable groups
...........................................
..................................................
.........................................................
98 104
....................................
a theorem
86
..............
....................................
interactions
specifications
INDEX
85
.........................
Gibbs
local
BIBLIOGRAPHY
71
the
quasi-invariant
transitions
71
the variational principle
A LATTICE
specifications
OF
........
function,and
Invariant
EQUIVALENCE
PRINCIPLE
...........................................
Local
DYNAMICAL
VARIATIONAL
..................................
a function
6.1.
7.1.
8.
AS
105 105 106 108 112 115
117 117 124 130 138
139
145
INTRODUCTION
It can be said that for the study of dynamical the crucial property of the acting group. classical
point of view is not only natural,
applications isometries
in statistical
systems amenability
This generalization
is
of the
but is also related
to the
mechanics where the acting group consists of
of the lattice.
This text, which grew out of a "third cycle" course in Ergodic Theory and Statistical the University
Mechanics
of Paris VI in 1980, deals with both topological
measure-theoretic dynamical
which I gave together with Didier Pinchon at
dynamical
systems,
systems of statistical
The existence of the ameaning
and
the symbolic
mechanics.
filter for an amenable group shall be
proved with the use of strongly dynamical
and in particular
subadditive
system of total orders.
Several
set functions
and the special
ergodic theorems
shall be
given. The entropy theory of measure-theoretic completely
described;
dynamical
and the Shannon-McMillan
corollary of a new ergodic
theorem,
systems shall be
theorem is given as a
the "almost subadditive
ergodic
theorem." A link between topological
and measure-theoretic
be made by way of the variational continuous
principle
function on a compact Hausdorff
dynamical
systems
shall
for the pressure of a space under the action of an
amenable group. A careful
study of subadditivity
use of tiling methods
in proving
of set functions several
allows us to avoid the
important
tiling is essential when proving the equivalence
theorems.
However,
of a countable
amenable
group with Z. This proof is given in the last chapter along with Rokhlin's lemma and the proof of the hyperfiniteness
of countable
(using the tower extension argument of Connes,
amenable groups
Feldman and Weiss).
Vl I would like to express my indebtedness significant portion of the material
to Didier Pinchon,
to whom a
contained in this text is due.
I also would like to thank Jean-Paul Thouvenot for many helpful discussions which stimulated the work on this text, and Tony Frank Paschall for his assistance with the English manuscript.
Jean Moulin Ollagnier Villetaneuse,
September 1984
i. P R E L I M I N A R Y ANALYSIS
I.I. S U B L I N E A R FUNCTIONS AND THE H A H N - B A N A C H THEOREM
I.I.I.
Definition.
Let E be a real vector
space. A real function p on E
is said to be sublinear if it is both subadditive and p o s i t i v e l y homogeneous,
i.e. if the two following conditions hold:
i)
for every pair
p(x+y)
(x,y) of vectors in E
ii) for every x and every positive number
1.1.2.
Remark.
~ p(x)
+ p(y)
p(Ix) = I p(x)
It is quite clear that a linear functional
is a sublinear
function and that the least upper bound of a set of sublinear functions, if one exists,
is still sublinear.
linear functionals
Therefore,
a least upper bound of
is a sublinear function. We shall state that this
p r o p e r t y is characteristic.
1.1.3. Example.
Consider the vector space C(X) of all real continuous
functions on the compact set X. Function s, defined on this space by the formula
s(f)
=
sup f(x) xeX
is sublinear.
1.1.4.
Extension theorem
(Hahn-Banach).
Let E be a real vector
space and
p be a sublinear function on E. Let F be a subspace of E and m be a linear functional b o u n d e d above by p on F, i.e. for every x in F, m(x) Then,
~ p(x).
there exists a linear functional n on the whole space E, still
b o u n d e d above by p, w h i c h is an extension of m. The proof of the theorem e s s e n t i a l l y depends on an extension lemma, and, w h e n the dimension is infinite,
on Zorn's lemma as well.
1.1.5.
Extension
lemma.
Let E be a real vector
function on E, G a vector functional
subspace of E non-equal
p a sublinear
to E, and f a linear
on G bounded by p.
Let a be an element of E\G so that the vector greater
space,
than G. It is then possible
a linear functional
space G 8 Ra is strictly
to find an extension of f which is
on G 8 Ra and is still bounded by p.
Proof of the lemma. We look for a real number ~ such that, in G and every real number ~, the following f(x) + ~ Using homogeneity,
< p(x+~a)
we have only to verify
f(x) + a
~
p(x+a)
and
that, f(x)
The real number ~ must then be chosen greater Sup yeG
for every x
inequality holds:
for every x in G,
- ~
~
p(x-a)
than or equal to
(f(y)-p(y-a))
and less than or equal
to
Inf (p(x+a)-f(x)) xeG Such a choice is possible f(y) - p(y-a) which is equivalent Because
to
if, for every pair <
p(x+a)
in G,
- f(x)
f(x) + f(y) ~ p(x+a)
f(x) + f(y) = f(x+y)
(x,y) of vectors
~ p(x+y)
+ p(y-a).
~ p(x+a)
+ p(y-a),
the proof is
achieved. 1.1.6.
Proof of the theorem.
G is a subspace
Consider
of E which contains
the set I of all pairs
(G,f) where
F, and where f is a linear functional
on G, is bounded by p, and extends m. This set I can be ordered in the following way: (G,f) $ (G',f')
<
Set I is non-empty because the order.
>
G
G' and f' is an extension of f
(F,m) belongs
to it and it is inductive
for
According
to Zorn's
for this order. lemma 1.1.5) 1.1.7.
For such a maximal
element,
element
(G+,n)
G + is equal to E (using
and the proof is thereby obtained.
Corollary.
linear functionals Proof.
lemma, we can then find a maximal
A sublinear
function is the least upper bound of all
less than or equal to it.
We want to show that,
for a given sublinear
for every vector x in a vector
function p on E and
space E, the real number p(x)
least upper bound of all numbers
is the
f(x), where f is a linear functional
on
E bounded above by p. Then,
the inequality
On the other hand,
f(x) ~ p(x) holds.
the linear functional
given by f(x) = p(x), whole
on the subspace generated by x,
can be extended according
to theorem 1.1.4 to the
space.
1.1.8.
Example.
describing
Continuing
now example
the linear functionals
for every continuous
1.1.3 of function
s on C(X) and
on C(X) bounded by s, we find that,
function f on X, a linear functional m bounded
above by s verifies m(f)
~ IIfII, where
II II is the uniform norm.
We then have m(f) ~ s(f) ~ IIfll
and
Therefore m is a Radon measure Moreover,
if f is everywhere m(-f)
On the other hand, m(1)
~ 1
and
$ p(-f)
m(-l)
$ -I
non-negative,
$ 0
and
measures
on X.
1.1.9.
~ 0
to constant
A pseudo-norm
the identity
is a sublinear
function which moreover
p(x) = p(-x)
of linear functionals
given pseudo-norm
p.
= 1
are bounded by s.
One of the more common forms of the Hahn-Banach extension
m(1)
equal to I, i.e. Radon probability
all Radon probability measures
Remark.
I, both inequalities
are true and therefore
on X with a total mass
verifies
m(f)
on C(X) bounded by s are then positive Radon
measures
Conversely,
$ s(-f) ~ IIfll
on X.
if f is equal
The linear functionals
-m(f) = m(-f)
theorem deals with the
whose absolute value
is bounded by a
If p is a pseudo-norm,
the two conditions m ~ p and
[m[ ~ p are equivalent
and theorem 1.1.4 enables us to complete the proof.
I.I.I0.
Remark.
It is possible to consider a similar problem with complex
rather than real vector spaces. In this case, a p s e u d o - n o r m p is a subadditive positive function on complex space E that verifies,
for every vector x and every complex
number ~,
p(~x) = I~Ip(x)
Then let m be a complex linear functional bounded by p on subspace F of E. According to remark 1.1.9,
the real
an extension v to the whole space,
linear functional on F, Re(m), has such that the absolute value of v is
b o u n d e d by p. The complex function n on E, given by n(x) = v(x)
- i.v(ix),
is a complex
linear functional with an absolute value bounded by p, and is an extension ofm.
1.2. COMPACT CONVEX SETS
1.2.1.
Definition.
Let E be a real vector space and E* its algebraic dual
space. Every x in E defines a p s e u d o - n o r m Px on E* in the following way:
px(f) =
f(x)
The family of all these pseudo-norms endows E* w i t h a topology and it becomes
a locally convex topological vector space.
This topology is the restriction to E* of the product
topology on R E .
It is the coarsest topology for which the coordinate applications from E* to R, f --> f(x),
are continuous.
It is called the weak* topology on E*. 1.2.2. Proposition.
Let K(E,p) be the subspace of E* that consists of
all linear functionals on E that are bounded above by the sublinear function p. K(E,p)
is a convex subset of E*, and it is compact for the weak* topology.
Proof. results
The convexity property is evident, from Tychonoff's
The product
space of all segments
RE; and the conditions
while the compactness
of K(E,p)
theorem. [-p(-x),p(x)]
defining K(E,p)
is a compact subset of
in this product
space make it a
closed subspace.
1.2.3. Examples. The unit ball of the strong dual of a normed space is weakly compact. The unit ball of the space of Radon measures w e a k l y compact.
on a compact
space X is
The subset of this ball of all probabilities
is convex
and weakly compact.
The following
lemma is useful
1.2.4.
Let F be a finite dimensional
Lemma.
in proving
the converse of proposition real vector
its dual space, and K a convex compact
subset of F'
Every linear functional
to K(E,p) when,
f(x)
Proof.
Consider
dian structure
~
f on F belongs
Sup geK
1.2.2.
space. Let F' be for every x in F,
(g(x))
a Euclidian
structure
< , >
on F and the dual Eucli-
on F'
Call f' the orthogonal
projection
of f on K, i.e.
the unique element
of
K such that Inf geK
Let x be the vector
=
such that
f - f' = <x,.>.
For every g in K and every real number e between 0 and I, belongs
(f'+e.(g-f'))
to K and (f-f')(x)
=< IIf-f'-e.(g-f')II 2
=
(f-f') (x) +
For every strictly positive 0
< 2(f'-g) (x) + EIIf'-gll2
2e. (f'-g) (x)
+
E2 llf'-gll2
When ~ tends to 0, this formula reduces to
(g-f')(x) ~ 0 , from which
we derive f'(x) = Sup g(x) geK Therefore f(x)
is less than or equal to f'(x), and the scalar square of
f-f' is non-positive, which means
1.2.5. Proposition.
that f belongs to K.
Let K be a convex part of the dual space E* of a
real vector space E, and let it be compact in the weak* topology. Let x be a vector in E, and consider the least upper bound p(x) of all numbers
f(x), where f belongs to K. For every x in E, p(x)
So defined, and K(E,p) Proof.
function p is sublinear,
is finite.
and the two compact convex sets K
are equal.
For every x, the function
therefore bounded.
f --> f(x)
is continuous on K and
Function p is then defined on the whole space E and
it is a supremum of linear functionals.
According to remark 1.1.2, p is
sublinear. By definition, K is contained in K(E,p). To prove the converse inclusion,
consider an element f of K(E,p).
For every finite dimensional vector subspace of E, there exists according to lemma 1.2.4 a convex compact n o n - e m p t y subset K F of K whose elements give the same value as f to all elements of F. The family of all compact sets K F has the non-empty finite intersection property.
Therefore,
there exists a convex compact subset K E of K, whose
elements give the same value as f to all elements of E. There is only one element in K, which is indeed f, and the converse inclusion is proven.
1.2.6. Remark.
It is possible to deduce the result of p r o p o s i t i o n 1.2.5
from a geometrical
form of the H a h n - B a n a c h theorem.
1.3. RADON MEASURES
1.3.1.
Definition.
Consider the vector space C(X) of all real continuous
functions on a given compact Hausdorff space X. A Radon measure on X is a linear functional on C(X),
continuous for the s u p r e m u m norm given by
the formula
1.3.2. gical
IIfll = Sup xeX
Definition.
If(x) l.
Let
space and ~ a
(X,~) be a measurable
o-algebra
A positive real measure measure
of all its compact
is said to be outer regular the greatest 1.3.3.
subsets.
space,
theorem.
and let A be a positive
Then there exists
a o-algebra ~ i n
~ represents
real measure on this space
of every measurable
set is
of all open sets containing
linear functional
on Cc(X),
on X with a compact
X which contains
all Borel
support. sets, and
the following properties:
A
feCc(X)
---> ~(f)
=
f f d~ X
second,
~ is inner and outer regular and gives a finite measure
compact
sets; and,
third,
the o-algebra ~ is complete
One proof of this can be found in Rudin 1.3.4.
Remark.
theorem,
the Borel o-algebra 1.3.5.
Corollary.
correspondence,
and also, when necessary,
the restriction
Let ~ be a Radon measure on a compact Hausdorff semi-continuous
Proof.
to
space
function on X. Then the integral
is the supremum of all ~(~), where ~ is a continuous
greater
we shall
derived from a linear functional
of this measure.
X, and f be an upper ~(f)
to
for ~.
(i).
Because of the above one-to-one
call a Radon measure both the measure on C(X) by Riesz's
it.
Let X be a locally compact
real functions
a unique positive measure ~ on ~ w i t h first,
A positive
if the measure
lower bound of the measures
the space of all continuous
if the
subset of X is the least upper bound of the
The Riesz representation
Hausdorff
the Borel o-algebra.
on this space is said to be inner regular
of every measurable
measures
containing
space, where X is a topolo-
function on X
than or equal to f. For every continuous
results
from the positivity
function ~, the inequality
~(f)
~
~(~)
of ~.
Consider now,
for every natural number n, the function fn which is equal
to Sup(f,-n),
and is so upper
Sequence
(fn) then decreases
theorem,
the integral
semi-continuous to f. According
and bounded. to the monotone
of f is the limit of the integral
convergence
of fn when n
tends to infinity. We have only then to prove the results for bounded upper semi-continuous functions.
By adding a constant we can even restrict the proof to the
case of non-negative functions. To achieve such a proof,
let g be a n o n - n e g a t i v e upper semi-continuous
function on X. For every positive real number 8, g~ is the function given by
= g~ In fact,
~
l{g~n6}
6"n= 0
there is a finite number of terms in the sum that are different
from 0 since g is bounded.
The last subscript is
n O = E(Sup(g)/6).
This function is greater than or equal to g, and this inequality holds:
~(g~ - g)
<
~.~(I)
Given a positive real number ~, select a ~ less than e/2~(1).
Since u is inner regular,
every compact set K n = {g~n~}
has an open
n e i g h b o r h o o d O n such that ~ ( O n - K n) < n Then,
there exists a continuous function 0n , whose values
lie between 0
and i, and which is equal to 1 at every point of K n and to 0 at every point outside of 0 n. The positive real number n is then an upper bound for
~(0 n) Choose n less than
~(I K ) n E/2 Sup(g)
and consider the function
no 0
=
5.( i + n~I= 0n )
This function is greater inequality
u(+-g)
< ~
than or equal to g at every point in X, and the holds, which concludes the proof.
1.3.6. A version of Dini's continuous
lemma. Let (fi)iel be a set of upper semi-
functions on a H a u s d o r f f compact set X such that every pair
of elements has a common lower bound in the set, i.e. that this set is directed for the order ~. Let f be the greatest
lower bound of all these functions.
The following m i n i m a x result holds in this situation:
Sup f(x) xeX
=
Inf Sup fi(x) ieI xeX
To demonstrate this consider first the obvious inequality
Sup f(x) ~ Inf Sup fi(x) xeX ieI xeX In order to prove the converse inequality denote by a the following number :
a = Inf Sup fi(x) ieI xeX If a is equal to -~ Otherwise,
, there is nothing to prove.
the compact sets
fil(La,+=L)
are n o n - e m p t y and their family
has the n o n - e m p t y intersection property. At every point x of the intersection of all these compact sets,
the
limit function f takes a values f(x) which is less than or equal to a; and this concludes the proof.
1.3.7. Corollary.
Given a convex and compact set of Radon p r o b a b i l i t y
m e a s u r e s on a compact Hausdorff set X, and an upper semi-continuous f u n c t i o n f on X, the function ~
--> ~(f)
on K is also upper semi-
continuous. Moreover,
Sup ~(f) ~eK where
=
Inf Sup ~(~) ~ f ~eK
the infimum is taken on the set of all continuous
functions on X
b o u n d e d below by f.
Proof.
Corollary 1.3.5 implies that the function ~ --> ~(f)
of a set of continuous
functions,
is an infimum
and thereby upper semi-continuous.
The m i n i m a x result is obtained by application of lemma 1.3.6.
10
1.4. EXTREMAL POINTS IN COMPACT CONVEX SETS
1.4.1.
Definition.
A point M in a convex part C of a real vector space E
is said to be an extremal point in C if it is not a convex c o m b i n a t i o n of two other points in C.
In the following extremal points of particular compact convex sets of p r o b a b i l i t y measures will be used.
The next property,
product of the K r e i n - M i l m a n theorem,
demonstrates
which is a by-
the existence of
extremal points.
1.4.2.
Proposition.
space,
and K a convex and compact subset of E.
The set Ext(K)
Let E be a locally convex Hausdorff real vector
of all extremal points in K is not empty.
Proof. Let I be the set of all closed non-empty subsets of K, for which the so-called open-segment p r o p e r t y holds:
every open segment ] A BF meets
(]A B E
= {=A+BB, ~+~=I,~
>0,B
>0})
that
C is contained in C.
Consider the converse order of the inclusion order on I; K being a compact set, exists,
I is hence an inductive p a r t i a l l y ordered set. Then,
according to Zorn's lemma,
there
some maximal element in I.
It remains to be seen that any such maximal element reduces to a single point, which would then be an extremal one in K. Let C be an element of I w h i c h is not reduced to a point.
Given two
different points x and y of C, there exists a continuous p s e u d o - n o r m p such that p(x-y)
is strictly positive because E is a H a u s d o r f f space.
The e x t e n s i o n theorem 1.1.4 enables us to build a continuous functional
f on E such that
f(x-y) = p(x-y)
> 0.
Denote by C + the subspace of C consisting of all points reaches
linear
z at which f
its m a x i m u m value on C.
Because C is compact,
C + is not empty.
It it clearly a convex and closed
subset of C. The open-segment p r o p e r t y still holds for C +. Indeed,
consider a convex combination
M = ~A + BB, where ~ and B are
n o n - n e g a t i v e and whose sum is equal to i; and, if M belongs to C +, A and B also belong to C because the open-segment property holds for C.
11
Moreover,
the inequalities f(A) ~ of(A) + ~f(B)
imply
f(A) = f(B) = f(M)
Therefore
1.4.3.
Example.
extremal
f(B) ~ of(A) + ~f(B)
, and the points
C + is an element
and this concludes
and
A and B then belong to C +.
of I, strictly greater
than C for the order;
the proof.
The following
situation will enable us to characterize
points.
X is a compact Hausdorff (defined in 1.1.3)
space and s is the sublinear
so that K(C(X),s)
all Radon probability measures N is a vector
function on C(X)
is the convex and compact
set of
on X (1.1.8).
subspace of C(X) such that,
for every f in N, s(f)
is non-
negative. The null functional
is then bounded by s on N and it is possible,
ing to theorem 1.1.4,
to find a functional
space E whose restriction The non-empty
to N is 0.
set K(C(X),s,N)
give an integral
equal
the extension p(f)
The following
=
on X that
of N is then a convex and
this compact and convex set is characterized
by its upper bound p according Applying
of all Radon probability measures
to 0 to all elements
compact part of K(C(X),s);
accord-
bounded by s on the whole
to 1.2.5.
lemma 1.1.5 to the space N • Rf, p may be written
Sup meK(C(X),s,N)
m(f)
=
Inf s(f+g) geN
theorem gives a characterization
of the extremal
points
in
K(C(X),p). 1.4.4.
Theorem.
In 1.4.3,
the extremal
the measures m for which the subspace
points N @ R.I
in K(C(X),p)
are precisely
is dense in C(X)
for the
p s e u d o - n o r m m(l.l). Proof. i) Let us first show that a measure with the above property Consider
a strict convex combination
property holds
for m:
(=i.o2 > 0) where
is extremal.
the density
12
m = ~iml + ~2m2 The two linear functionals m I and m 2 are continuous m(I.l)
for the p s e u d o - n o r m
since
Iml(f) l
~
(1/~ 1) m(Ifl)
Im2(f) l ~ (1/~ 2) m(Ifl)
and
They are equal to m on the dense subspace N @ R.I, and therefore are equal on the whole space. Functional m is then an extremal point of K(C(X),p). 2) The converse can be proven by contraposition. If
N ~ R.I
is not a dense subspace of C(X) for the p s e u d o - n o r m m(l.l),
there exist a real continuous
function f and a strictly positive real
number ~ such that, for every g in N and every real number a,
m(l f-g-al )
>
~
>
0
Consider the linear functional 8 on
0(g + a + ~f) =
N • R.I @ R.f
given by
~=
The functional verifies on this subspace the inequality
e ~ m(I.I).
It is then possible to find on the whole space C(X) a linear functional b o u n d e d above by m(I.I) , whose restriction is 0. Denote it by ~. The d e c o m p o s i t i o n
m
=
(1/2)
((m+~) + (m-~))
is non-trivial because #(f) It remains to be shown that
is not 0. m+~
and
m-~
are elements of K(C(X),p);
but the only thing to be proven is that they are positive measures. If h is a continuous non-negative
(m + ~¢)(h)
1o4.5. Remark.
=
m(h)
The dense subspace
function on X, we get (s = jl)
+ E¢(h)
N @ R.I
>
m(lh[)
- [#(h)[
>
0
is then dense in LI(X,Q,m)
the L l - n o r m , where m is the regular measure built from the functional m by theorem 1.3.4. 1.4.6. Remark.
Compact convex sets described in 1.4.3 are simplexes and
for
13
all compact 1.4.7.
simplexes
Example.
of probability measures have such forms.
Let f be a continuous
mapping
from a compact
space Y
onto a compact set X, and let m be a Radon probability measure The set of all probability measures type described Extremal morphism
in this convex set are exactly those for which f is a
of the two measure of the quotient
the o-ideals 1.4.8.
in 1.4.3.
points
conjugacy
spaces
(Y,~,n)
Boole algebras
and
(X,~,m),
i.e. an iso-
of the o-algebras
of events by
of null sets.
Example.
Let X be a compact
A Radon probability measure
set and T a h o m e o m o r p h i s m
of X.
~ on X is said to be invariant under the
action of T if the equality function
on X.
on Y whose image by f is m is of the
~(f)
= ~(foT)
holds
for every continuous
f on X.
The set M(X,T)
of all Radon probability measures
on X, which are invar-
iant under T, is not empty and is also a convex and compact set of the type described Extremal invariant
elements
Let us prove Here,
in 1.4.3.
invariant probabilities of LI(X,~,~)
are ergodic, are constant
which is to say the only functions.
these results.
the space N is generated by the increments
belongs
to C(X).
increment It shall
Every linear
f - foT
combination ~ ~i(fi-fioT)
where
f
is still an
f - foT. suffice to verify
s(f-foT)
~ 0
Let x be a point in X at which f reaches a point the difference therefore,
(f-foT)(x)
s is non-negative
The extremal
points
in M(X,T)
its least upper bound.
is greater
At such
than or equal to 0, and,
on N. are those for which the subspace N @ R.I
is dense in LI(X,~,~ ). Consider
now the sequence of contraction
operators
A n of LI(X,~,~)
where
A n is given by the expression An(f) The sequence
=
(An(f))
n-I (I/n). ~ loT i of averages
converges
to the expectation
~(f)
for
every element of N • R.I; and, by virtue of the density of this subspace, the convergence
result also holds
for every element of LI(X,~,~).
14
If f is an invariant to u(f).
An invariant
if u is extremal 1.4.9.
element,
Remark.
An(f)
is a constant
element of LI(X,Q,u)
In the previous
space.
In Chapter
that converges
is then a constant
function
in M(X,T). example we proved that there always
at least one invariant probability under Hausdorff
sequence
This was demonstrated
2, we shall examine
a homeomorphism
by Bogoliubov
the transformation
are invariant probability measures whenever
and Krylov
groups
exists
of a compact (1).
for which there
they act on a compact
space
by homeomorphisms. To prove the existence we used a convergence ergodic
theorem.
of invariant probabilities theorem of means,
in the present
which is a special
chapter,
case of a mean
Chapt,er 3 deals further with this topic.
1.5 REFERENCES
For further Bourbaki's
study of Hahn-Banach text devoted
Riesz representation Choquet
theorem,
to topological
refer to Meyer
vector
spaces
theorem is proved by Rudin
(i) is recommended
for general
and more in-depth study of simplexes.
(i) or to
(3).
(i, chapter
information
2).
on functional
analysis
2. DYNAMICAL
2.1. DYNAMICAL
2.1.1.
SYSTEMS AND AMENABLE GROUPS
SYSTEMS
Definition.
We call a pair
when X is a compact Hausdorff
(X,G)
a topological
dynamical
system
space and G a group of homeomorphisms
of
this space. When discussing
the action of an abstract group G, we shall denote by T g
the homeomorphism (X,T)
of X associated with the element g of G and denote by
the corresponding
dynamical
system.
When G is the group Z of all relative the generator
integers,
T is simply the image of
i.
2.1.2. Example.
The first example
set, and X the product
is given by shifts.
space I G of all mappings
Endowed with the product topology of the discrete a compact Hausdorff
space which is metrizable
Given an element g of G, consider Tg(~)(a) The mappings
topologies,
X becomes
when G is countable.
the mapping T g from X to X:
= ~ (ag)
T g are homeomorphisms
group of all homeomorphisms corresponding
Let I be a finite
from G to I.
of X; and the mapping from G to the
of K, obtained by :sending every g to the
T g, is a group homomorphism.
Such a group homomorphism We shall sometimes
is called an action of G on X by homeomorphisms.
refer to T g as a translation.
Indeed,
T g is the right
translation by g-I of the graphs r($): (a.g-l,i) The set I is sometimes we have constructed 2.1.3.
Example.
e
r(Tg(¢))
<
>
(a,i)
called an alphabet,
is called a symbolic
Here is another example,
e
r(~)
and the dynamical
dynamical
system that
system.
which is as symbolic as the
16
preceding.
Given a group G, consider
the set T(G) of all total orders
on G. If t is a total order on a finite part F of G, O(F,t) the set of all total orders on G whose restriction The set of all O(F,t)
is one of the bases of a topology on T(G).
Endowed with this topology, metrizable to T(G)
T(G)
when G is countable;
of the product
Of course,
topology
is a compact Hausdorff on
<==>
x < y
For every element g in G, consider Tg(r)
(a,b)
=
The T g are homeomorphisms
2.1.4.
As in example
Definition.
of T(G)
the mapping
and mapping
is a one-to-one
An automorphism
bimeasurable mapping
Definition.
to T(G):
are called translations.
if T and its inverse
of a probability
space
from X to X that preserves
i.e. for every A in the u-algebra (T -I(A))
from T(G)
g to T g gives an action of G
A mapping T is bimeasurable
are both measurable.
the measure,
for
2.1.2 these homeomorphisms
(X,~,~)
4, the following holds:
= ~(A)
Given a probability
action of G on (X,~,~) system.
{0,I} GxG.
~(ag,bg)
mapping
2.1.5.
space, which is
this topology is simply the restriction
we identify an order T with its graph: T (x,y)=l
on T(G).
stands for
to F is t.
space
by automorphisms
(X,~,~)
and a group G, an
is called an abstract
Such an action is then a family of automorphisms
lity space, which is indexed by the elements
dynamical
of the probabi-
of G, and for which the
equality T gh
holds 2.1.6.
=
for every pair Remark.
TgoT h
(g,h) of elements
The question of modulo
spaces will be investigated In these first chapters, pological
of G. 0 automorphisms
and Lebesgue
later.
abstract
dynamical
ones and the automorphisms
systems are derived from to-
are therefore well-defined
mappings.
17
2.1.7.
Proposition.
Let
(X,G) be a topological
be a Radon probability measure of G. This means
every element g of G, the following
Then,
=
function f on X, and for
equality holds:
of the probability
space
is the o - a l g e b r a built by the Riesz representation
lar,
Indeed,
and
~ (f)
the T g are automorphisms
Proof.
system,
on X, which is invariant under the action
that for every continuous
u (foT g)
dynamical
the T g are measurable
for ~; and,
(X,~,~) where
theorem
(1.3.3).
since ~ is inner regu-
for every A in Q, (A)
Of course, compact
=
Sup KcA
~ (K)
here we take the least upper bound of the measures
of all
subsets of A.
A similar result holds for (Tg)-I(A). On the other hand,
according
to corollary
the compact set K, is the greatest continuous Then,
functions
1.3.5, ~(K),
the measure
lower bound of the measures
above the indicator
of
of all
of K.
for every element A in the o-algebra
~, the following
equality
holds: ~(A)
2.1.8.
Example.
=
~((Tg)-I(A))
Taking the topological
dynamical
duced in 2.1.2 we can define a probability Let
system that we intro-
on it in the following way.
(Pl .... ,pn ) be a n-uple of strictly positive
is I, where n is the number of elements If f is a function only depending example
the coordinates
(f) If we take another
=
real numbers whose
sum
in I.
on a finite number of coordinates
in the finite part F), ~(f)
(for
is defined by
i=n ~ F f(a).( ~ Pa. ) aeI i=l l finite part F' that contains
F, the value of u(f)
is
the same. Then the above formula defines value
1 to the constant
a positive
function
linear functional,
I, on the vector
giving the
space of all functions
that only depend on a finite number of coordinates.
18
According
to the Stone-Weierstrass
the uniform norm on C(X), can be extended
theorem,
this subspace
and the uniformly continuous
to the whole
space C(X).
is dense for
linear functional
Here we get a Radon probability
measure ~ on X. It is easy to see that ~ is invariant under tions.
Such an abstract
2.1.9.
Example.
dynamical
Consider
the action of G by transla-
system is called a Bernoulli
the dynamical
system T(G)
introduced
scheme. in 2.1.3
of all total orders on G. All O(F,t)
are open and closed
Let us define a positive which
consists
subsets
of T(G).
linear functional ~ on the subspace of C(T(G)),
of all functions
depending
solely on the restriction
of
the order to a finite part of G, by giving in a coherent way the mass of all O(F,t). It is possible
to do this by setting
The above subspace
~(O(F,t))
is dense in C(T(G))
the Stone-Weierstrass
theorem;
for the uniform norm according
and the functional
of a unique Radon probability measure still
= I/IFI! to
~ is the restriction
on T(G) which we shall nonetheless
call simply ~.
One can easily ensure that ~ is invariant under the action of G on T(G) by translations;
in fact, ~ is invariant under the action of the wider
group of all homeomorphisms According
of T(G)
2.1.10.
Remark.
variant
Radon probability measures
systems.
In the two previous
However,
probability
topological
to find invariant
topological
dynamical
2.1.12.
Radon
system.
for every action on a compact Haus-
there is at least one invariant
A group G is said to have the fixed-point
acting by affine continuous
one-to-one mappings
pact and convex subset of a locally convex Hausdorff space,
dynamical
Radon
As we saw in 1.4.9 the group Z has this property.
Definition.
if, whenever
of G.
system.
examples we were able to build in-
in an arbitrary
dorff space by homeomorphisms, probability.
dynamical
in particular
it is not always possible
measures
But there exist groups such that,
2.1.11.
induced by the permutations
to 2.1.7, we then get an abstract
this compact convex space contains The Markov-Kakutani
theorem.
property on a com-
topological
vector
at least one invariant point.
An Abelian group has the fixed-
point property. The study done in 2.3 leads to one proof of this result,
cf. 2.3.6.
19
2.1.13.
Remark.
Concerning vocabulary:
the study of dynamical
systems
b e g a n with some actions of R and Z, which represented r e s p e c t i v e l y the e v o l u t i o n of a mechanical
system through time, and the d i s c r e t i z a t i o n
b e t w e e n regular time intervals of this evolution; hence the adjective "dynamical". In this text the most significant examples of dynamical be found in statistical mechanics.
systems shall
In these examples the group G does
not connote evolution but consists of isometries.
However,
not prevent us from calling such an action a dynamical
that will
system.
2.2. THE FIXED POINT PROPERTY AND THE AMEANING FILTER
2.2.1 Definition. We call a group G amenable if it has the fixed-point p r o p e r t y described in 2.1.11, however used, 2.2.2.
cf. Greenleaf Definition.
several other definitions
can be
(i). A group is said to give an invariant version of the
H a h n - B a n a c h theorem if the following extension result holds: Let E be a real vector space,
s a sublinear function on E, and T a
right action of a group G on E by linear one-to-one mappings preserving s, i.e. for every f in E and every g in G,
s(f) = s(f.Tg).
Let
F be a subspace of E invariant under the action of G, and m a linear functional on F bounded by s and invariant under the action of G. There exists a linear functional on E bounded by s and invariant under the action of G whose r e s t r i c t i o n to F is m.
2.2.3.
Proposition.
A group G gives an invariant version of the Hahn-
Banach theorem if and only if it has the fixed-point property.
Proof. Let us first show that the fixed-point p r o p e r t y leads to an invariant version of the H a h n - B a n a c h theorem. A c c o r d i n g to the H a h n - B a n a c h theorem, cribed in 2.2.2,
there exist,
in the s i t u a t i o n des-
linear functionals on E bounded by s, whose r e s t r i c t i o n
to F is m. The set of all these linear functionals of E* for the weak* topology.
is a convex and compact subset
20
The group G acts on this set by the affine one-to-one mappings
Tg(n)(f)
=
given by
n(f.T g)
There is an invariant point in this compact and convex set, i.e. a linear functional bounded by s and invariant under the action of G on E whose r e s t r i c t i o n to F is m.
In order to demonstrate the converse,
let K be a convex and compact
subspace of a locally convex Hausdorff topological vector space E, and let T be an action of a group G on K by affine continuous one-to-one mappings. Let E be the real vector space of all continuous real functions on K, and s the usual
sublinear function on E (defined in 1.1.3).
From the left action of G on K we can deduce a right action of G on E and still denote it by T
(f.Tg)(x)
=
f(Tg(x))
Next we take the subspace reduced to the function 0 as the subspace F. Thanks
to the fixed point property,
there exists a Radon p r o b a b i l i t y
m e a s u r e on K invariant under the given action of G. Because K is a compact and convex part of E*, the center of gravity of this p r o b a b i l i t y is an invariant point of K.
2.2.4.
Example.
We can prove that finite groups are amenable.
Indeed,
if
the finite group F acts on the compact and convex set K, the m e a n value
(IlIFI). [ Tg(x) geF is a fixed point of this action for every x in K.
2.2.5.
Example.
To prove that Z is amenable,
let K be a compact convex
set and T be an affine continuous one-to-one mapping of K. Consider a point x in K and the sequence
(Mn) of affine continuous oper-
ators on K given by
Mn
=
n-i (l/n). ~ Ti
Let x' be a limit point of the sequence
x n = Mn(X).
And let p be any of the continuous pseudo-norms
that define the topology
21 of the locally convex topological
vector
space in which K lies.
Because p is continuous
p(x'-T(x'))
~
Inf Sup P(Mn(X)-T.Mn(X))
N And because
Mn(X)
- T.Mn(X)
p(x'-T(x'))
= (i/n).(x - Tn(x))
~ Inf Sup (2/n).p(x)
N The value p(x'-T(x'))
n~N
n~N
is then equal to 0 for every continuous
n o r m on E; and because E is a Hausdorff
space this means
pseudo-
that x' is an
invariant point. 2.2.6.
Remark.
property
In the above example,
we demonstrated
the fixed-point
for Z by finding a sequence of finite parts of Z, the segments
A n = {0 ..... n-l},
such that the ratio
(I/IAnl).IAnAAn T I tends to 0
when n tends to infinity. All of which 2.2.7. F(G),
leads us to the following
Definition.
definition.
Given a finite part D of G, m D is the function on
the set of all finite parts of G, defined by mD(A )
=
l{xeA,~deD,dx~A}l
Intuitively we can see that A is as invariant under the right translations by the elements 2.2.8.
Definitions.
of D as the ratio
mD(A)/IA I
is small.
A group G is said to have an ameaning
filter
if, for
every finite part D of G and every positive real number 6, there exist finite non-empty parts of G such that the ratio
mD(A)/IA 1
is less
than o. The non-empty
set of all these finite parts of G is then denoted by
M(D,6 ). It is clear that if D' contains contained
The parts M(D,~) positive
of F(G)\{~}
real number)
we call the ameaning 2.2.9.
Proposition.
amenable.
D, and if 6' is less than 6, M(D',6')
is
in M(D,~). (where D is a finite part of G, and 6 is a
are then a basis of a filter M on F(G)\{~},
which
filter of G. If the group G has an ameaning filter,
it is then
22 This proof shall be similar
to the one in example
2.2.5.
Let K be a compact and convex subset of a locally convex Hausdorff
topolo-
gical vector space, and T an action of the group G on K by affine continuous one-to-one mappings. For every finite and non-empty part of G, consider
the average operator
M A given by MA
=
For every pair
(I/IAI).
(D,~),
~ Tg geA
F(D,~)
stands for the closed subset of K, which is
the closure of the union of all images of K by the operators
M A where A
is an element of M(D,~). Because
the set of all M(D,~)
has the non-empty
is a filter basis,
finite intersection
Next let F be the common intersection
the family of the F(D,~)
property. of all F(D,~),
and let y be a point
in F. For every h in G and every continuous
pseudo-norm
p on E, the following
inequality holds: p(y - Th(y))
Then,
Inf (D,~)
Sup AeM(D,~)
Inf (D,d)
Sup AeM(D,~)
P(MA(X)
Th.MA(X))
(2/IAl).m{h}(A).p(x)
for every h in G and every p,
p(y - Th(Y))
= 0
And y is invariant under the action of G. The rest of part 2.2 is devoted to the proof of the converse of the previous result:
every amenable group has an ameaning
This theorem was first proved by E. F~Iner tained in our work 2.2.10.
filter.
(I). We give here the proof ob-
(2) which comes from the study of invariant
Definition.
An invariant
capacity
capacities.
is a real function on F(G) with
the four following properties: i)
m(~)
=
0
ii)
for every pair
(A,B),
(strong subadditivity)
m(AUB)
+ m(A~B)
~ m(A) + m(B)
23 iii)
for every finite part A and every g in G,
iv)
there exists
m(A)
= m(Ag)
(right invariance) a positive
and a, the increment
2.2.11.
Example.
the cardinal
A -->
functions m D for instance, 2.2.12.
m(A~{a})
The first example
function
Proposition.
constant K such that,
of an invariant
Properties
used to verify
than -K.
capacity is given by
as the following
proposition
examples,
shows.
For every finite part D of G, the real function m D
(i) and
(iii)
capacity.
evidently hold.
The constant
I can be
(iv).
Next we prove the decisive Therefore,
is greater
IAI. There are some less trivial
on F(G) defined in 2.2.7 is an invariant Proof.
- m(A)
for every A
calculate
strong
subadditivity
the difference
which is the integral
mD(A)
property
+ mD(B)
for the counting measure
(iii).
mD(AnB)
- mD(AUB),
on AUB of the function
IA.I(D-IAC ) + IB.I(D-IB c) - !(ANB).I(D-I(A~B)C ) - I(D-I(AuB)c ) This function
is always non-negative,
dering every possibility, 2.2.13.
Definition.
greatest
The mean value of an invariant
capacity.
capacity m is the
lower bound q(m) of the ratio m(A)/IA I where A is a finite non-
empty part of G. Due to the property ties,
as one can see by carefully consi-
and thus m D is an invariant
q(m)
2.2.14.
Remark.
whereas
the function
homogeneous
(2.2.10
(iv)) of invariant
capaci-
is finite. The set ~ of all invariant m-->
q(m)
capacities
is a convex cone,
is clearly increasing
and positively
on E.
The existence
of the ameaning
filter
is proven when,
for every finite
part D, the mean value q(m D) is equal to 0. It is clear that the following
mD ~ Therefore, point,
d~D
subadditivity
result holds:
m{d}
we must now state the result for the parts reduced to a
and show the subadditivity
property of q on the cone E.
24
Let then d be an element of G. If d generates Otherwise,
a finite group F, one has
when n tends to infinity,
(i/IFl).m{d}(F)
= 0 .
the limit of the ratio
(i/I Anl ) .m{ d} (An) where
A n = {di,i=O,..,n-l},
The subadditivity
is equal to O.
property of q is the most important point of the proof.
In order to demonstrate
it we will rely on several
But first we must state a definition 2.2.15.
Definition.
support,
Let f be a non-negative
tive coefficients,
of f as a combination
we select a special
0 = s 0 < ~i < "'" < o k
values of f in ascending The following
special
2.2.16.
of indicators
one as follows.
be the finite sequence of the different
i=k ~ (~i-~i_l) I i= 1 " (f>=i) of f is called its pyramidal
Let m be a strongly
subadditive
decomposition.
function on F(G),
and f
function with a finite support on G.
Among all possible decompositions with positive
coefficients,
est lower bound to the sum Consider
f =
of f as a combination
the pyramidal
decomposition
of indicators gives the great-
[ ~Am(A).
an arbitrary
decomposition
of f:
~ ~AIA Ael
Let J be the set of all finite parts of G which is obtained by making repeated unions
and intersections
Then J is finite and the finite parts Choose a maximal of elements
(f~i)
element among all finite
in J containing
Every indicator were not,
with posi-
order.
decomposition
Lemma.
a positive
Proof.
with a finite
equality holds:
f = This
function,
lemma.
on a set G.
Among all decompositions Let
theorems.
and prove a combinatorial
the
I A is constant
(f~i),
of elements
from I
of I.
belong to it.
strictly increasing
and call this sequence
on every non-empty
the finite part K' = (AnKi+I)UK i
set Ki+ ~ K i ;
sequences (KI,.,Kn). if it
would be strictly between K i
25
and Ki+ 1 for the order, and this would contradict the maximality of the sequence. We use an Abel transform to obtain
(~i-ai_l) m(f>~ i)
=
~km(f=~k) +
k-I ~ ~i(m(f_->~i) - m(f>~i+l)) 1
Inserting the other K i of the sequence, we see that the first sum is equal to n f(i) .(m(Ki)-m(Ki_l))
+
f(1) .m(K I)
where f(i) is the constant value of f on Ki\Ki_ 1 (on K 1 for f(1)). Replace f by the given decomposition to get the following expression of the sum relative to the pyramidal decomposition: n
A~ei ~A
~ (A(i)'(m(Ki)-m(Ki-l))
+ A(1)'m~KI)
)
Because of the strong subadditivity property of m, the difference m(Ki)-m(Ki_l) equal to i.
is less than or equal to
m(A~Ki)-m(A~Ki_l) , when A(i) is
We then get the inequality
(~i_~i_l).m(f>~i)
<
~ ~A.m(A) Ael
which is the result sought to accomplish the proof. 2.2.17. Corollary. Mapping a function f with a finite support on G to the real± number ~ (~i- ~ i - l ) ' m ( f ~ i ) gives a sublinear function on the cone C$(G), which itself consists of all non-negative functions with a finite support on G. The restriction of this mapping to the indicators is the capacity m. The correspondence between m and its sublinear extension is a linear mapping. The proof of all these results is straightforward. 2.2.18. Definition. Two functions fl and f2 on G are said to have the same variation table when, for every pair (x,y) of elements in G, the following inequality holds:
26
(fl(x)
- fl(y)
).(f2(x)
- f2(y)
belong
to C~(G),
they have the same variation
If the two functions
if and only if the inclusion of all
(fl~=i)
2.2.19.
Proposition.
additive Proof.
The canonical
(K 1 ..... Kn)
to an invariant with a finite
extension
sublinear
to m(fl)
function
The next
For every element This
- m(g),
f in CK(G)
where
invariant
Proof.
to
(fl~i)
and all
in the proof of lemma 2.2.16
invariant
on the whole
function
+ on the cone CK(G)
space CK(G)
of functions
extension
of an invariant
capacity
the least upper bound of the differences g and f+g belong
on CK(G)
and an extension
To get a non-negative
+ f-, where belongs
is
lemma deals with this.
the functions
least upper bound defines
near,
of all
capacity
table.
+ m(f2).
2.2.20. Lemma. Let m be the canonical + to the cone CK(G). m(f+g)
sequence
of m(f)
this sublinear
support.
table
set consisting
of an invariant
have the same variation
the increasing
is equal
We have now to extend
0
is a total order on the finite
The second expression
that m(fl+f2)
~
(f2~j).
when the two functions
Call
(f2~Bj). shows
and all
)
a real
to C~(G),
function which
of m.
sum f+g we have to choose
f- is the negative
C~(G).
is finite. is subli-
a g equal
to
part of f (f = f+ - f-), and where
We then look for the least upper bound of all differences m(f + + ~)
m(f- + ~)
+ with a function ~ in CK(G). The following relation holds: m(f++~)
- m(f-+})
=
(m(f++#)-m(})) m(f +)
And property
(iv) of 2.2.10
h is the counting measure:
easily gives
+
+ (m(#)-m(f-+%))
(m(+)-m(f-+~))
the following
inequality
where
27 m(f-+~)
- m(~)
The difference
m(f+g)
real function
on CK(G)
- m(g)
still denote
-K.h(f-) is then bounded
is therefore
Because m is subadditive shall
>
extends m, but for simplicity
on CK(G) , function m is invariant to be shown that m is subadditive.
of G containing
both supports
positive
c such that
of g and f+g. f + c.l A
c, the subadditivity
m(f+g+c.l A)
~
and positively It is possible
is positive. + of m on CK(G)
c.l A has the same variation 2.1.19
m(f+g)
m(g)
fl + c.l A
~
~
table as f+g, enabling
us
m(f+c.l A) - m(c.l A)
f2 + C'IA
m(fl+ f2 )
yields
to get the inequality
real number
and
to find a
m(f+c.l A) + m(g)
to use proposition
For every positive
homogeneous.
Let A be a finite part
On the other hand,
that
we
it by m.
It remains
number
and the
well defined.
this function
So defined
For such a number
by m(f +) + K.h(f-);
m(fl+
E, we can then find a function are non-negative f2 + 2c.I A)
c.l A such
and
2m(c.l A) +
Then m(fl+
f2 )
$
e + (m(fl+ C.IA)-c.m(A))
+ (m(f2+ C.iA)-c.m(A))
E + m(f I) + m(f 2) Because
this is true for every e, m is subadditive
2.2.21.
Corollary.
cone E, consisting linear
invariant
2.2.22.
Theorem.
The above extension of all invariant
functions
the right
capacities,
on the vector
from the
to the cone of all sub-
space CK(G). space of all real functions
on the group G.
Let m and n be two sublinear under
gives a linear mapping
Let E be the real vector
with a finite support
on CK(G).
translation
functions
on E, and let them be invariant
action of G on E given by
28
(f.Tg)(x) Then,
=
f(x.g -I)
for every element f of E, the inequality holds Sup ~eK(E,m+n,G)
~(f)
~
Sup ~(f) ~eK(E,m,G)
+
Sup ~(f) ~eK(E,n,G)
where the symbol K( ..... ) denotes
the set of all linear functionals:
first,
bounded by a sublinear
on a vector
finally, Proof.
space;
second,
function;
and,
invariant under a group action.
Consider
the product vector space ExE, the component-wise
action of G on it, and the sublinear
right
function u given by
u(fl,f 2) = m(fl) + n(f 2) The diagonal mapping
subspace A of ExE can be identified with E and this natural
sends u to m+n, yielding
the equalities
Sup ~(f) ~eK(E,m+n,G)
Sup ~eK(A,u,G)
=
Sup ~(f) ~eK(E,m,G) Sup ~(f) ~eK(E,n,G) Because of the fixed-point
=
Sup ~eK(Ex{0},u,G)
~ (f,0))
Sup ~eK({ 0}xE,u,G)
~((0,f))
property,
an invariant
subspace which is bounded by an invariant variant extension sublinear
~((f,f))
linear functional
sublinear
function has an in-
to the whole space which is still bounded by the given
function.
The least upper bounds in the right sides of the three equalities therefore
relative
Subadditivity
are
to the same set K(ExE,u,G).
immediately
results because
(f,f)
is the sum of (f,0) and
(0,f). 2.2.23.
on a
Theorem.
As a real function on z, q is subadditive.
Proof. To derive this result from theorem 2.2.22,
it will suffice to
identify q(m) with the least upper bound of ~(l{e }) on K(E,m,G). For every ~ of K(E,m,G), ~(l{e })
=
the following holds: (I/IAI).~(IA)
~
(I/IAl).m(A)
29
Whence the inequality q(m)
(I{ e} )
It remains to be shown that the linear functional ~ on E, given by ~(l{e }) = q(m) is bounded by m. For every f in CK(E) , using the counting measure h on G, we get ~(f)
=
h(f+).q(m)
-
h(f-).q(m)
Using the pyramidal decompostion of f+, we write h(f+).q(m)
=
~ (~i-~i_l).h(f~i)-q(m)
and then obtain the inequality h(f+).q(m)
~
~ ( ~ i - ~ i _ l ) . m ( f ~ i)
Since there exist invariant linear functionals
=
m(f +)
on CK(G) bounded by m,
the following inequality holds: Sup feC~(G),f~0
( - m(-f)/h(f)
)
< =
(m) q
The number ~(f) is therefore
less than or equal to
When c.l A tends to infinity,
this sum is the limit of
m(f +) + m(c. IA-f-) For an A sufficiently
large,
m(f +) + m(-f-).
m(c.l A) c.l A - f-
and f+ have the same variation
table. The function m then verifies m(f)
=
m(f +) + m(-f-)
which achieves the proof. 2.2.24. Remark. The fixed-point prove,
property was used for a second time to
for a negative f, the inequality
30 h(f)
q(m)
<
m(f)
This was merely for simplicity 2.2.25.
Conclusion.
Amenable
(see Moulin Ollagnier
groups have an ameaning
and Pinchon filter.
As stated in remark 2.2.14,
the subadditivity
on z remained
this has just been accomplished
to be proven;
(2)).
property of q as a function in the above
theorem.
2.3. AMENABILITY
AND ALGEBRAIC
CONSTRUCTIONS
We give now some results on the stability under
several
Amenability and direct 2.3.1.
algebraic
of the fixed-point
of abelian and solvable groups, limits,
follows
Proposition.
of their finite extensions
from this study.
If the group G has the fixed-point property,
if H is the image of G by a group h o m o m o r p h i s m Proof.
property
constructions.
and
s, H is also amenable.
Let T be an action of H on a convex and compact part of a locally
convex Hausdorff vector
space by affine continuous
one-to-one mappings.
Define an action of G on this convex set by setting Tg(x)
=
Ts(g)(x)
Because G has the fixed-point iant point under
property,
there exists
at least one invar-
this action of G, which is then an invariant
point under
the given action of H. 2.3.2.
Proposition.
A subgroup
Proof.
We use here the ameaning
of an amenable group is amenable. filter.
For every finite part D of H and every positive real number exists
a finite part A of G such that
roD(A)
=<
~. ]A[
5, there
31
Dividing A between
the left cosets of H in G that it meets,
we see that
this set A is the disjoint union of the Ai.x i = HxinA. On the other hand,
the set
{xeA,~deD,dx~A}
is the disjoint union of
the sets {xeAi,~deD,dx~A i} because D is a subset of H. The two equalities
IAI
therefore
simultaneously
= ~ IAil
and
At least one of the Aix i belongs
hold
mD(A)
to M(D,~)
=
mD(Aixi )
as well as its right trans-
late Ai, which is a part of H. We have now proven the existence 2.3.3.
Proposition.
amenable Proof.
of the ameaning
filter
for H.
Let G be an extension of an amenable
group N by an
group H. The group G is then amenable.
Let T be an action of G on K with the usual
The restriction
of this action
has some invariant points The non-empty
to the normal
subgroup N of G
in K.
subset K N consisting
convex and compact.
conditions.
amenable
Moreover,
of G because N is a normal
of all these invariant points
KIN is globally invariant under
is
the action
subgroup: -i
Tn(Tg(x))
The restriction
=
Tn'g(×)
=
Tg(T g
"n'g(x))
=
Tg(x)
to K N of the given action of G is in fact an action of
the quotient group H, i.e. the mapping T g, restricted
to KN, depends
solely on the coset of N to Which g belongs. Indeed,
if x belongs Tg-n(x)
to KN, =
Because H is amenable,
Tg(Tn(x))
Proposition.
nable groups, Proof.
Tg(x)
there is at least one invariant point in K N for
this action of H; this point 2.3.4.
=
is invariant
for the given action of G.
If G is the union of a directed family
(G i) of ame-
G is amenable.
The family
(Ki) of compact
convex subsets
of K, where K i is the
set of all points of K which are invariant under the action of Gi, has the non-empty
finite intersection
The intersection
property.
of all these K i is the non-empty
set of the points
that
32
are invariant under the whole action of G. 2.3.5.
Proposition.
Proof.
Consider
the usual
The free group with two generators
the vector
sublinear
the vector
is not amenable.
space of all bounded real functions
function
s and the action of L(a,b)
on L(a,b),
preserving
s on
space: f.Tg(x)
=
f(gx)
Let us show that there is no linear functional
on E both bounded by s
and invariant under the action of L(a,b). A linear functional
bounded by s is a positive
linear functional
with a
total mass equal to i. Call it a mean. Consider
the four following
subsets
of L(a,b):
A+
is the subset of all words beginning with a
A-
is the subset of all words beginning with a -I
B+
is the subset of all words beginning with b
B-
is the subset of all words beginning with b -I
The whole group L(a,b) of the part reduced
is the disjoint union of these four subsets
to the unit element.
For every mean on L(a,b), I
=
Moreover,
and
the equality holds
v(A +)
+
v(A-)
IA+oT a
~
1
+
~(B +)
+ v(B-)
+
v({e})
IA-
An invariant mean would then verify
~((IA+) which contradicts
+ (IA-))
~
I
and
~((IB+)
+ (IB-))
~
i
the above equality.
2.3.6.
Remark.
groups
and Z by using the constructions
All known amenable
groups are obtained
from the finite
that we have so far described
in 2.3. 2.3.7.
Conjecture.
A standard conjecture
is that a non-amenable
group
33
has a subgroup 2.3.8.
isomorphic
Proposition.
to L(a,b).
Consider
the group B, consisting
of Z with a finite support. the semi-direct product
The group G of permutations
of Z obtained
of B by the group of all translations
nable and finitely generated. solvable
of all permutations
However,
as
is ame-
G is not a finite extension of a
group.
Proof. Denote by S A the finite subgroup of all permutations
of the fi-
nite part A of Z. The group B is the direct Then,
according
limit of S A when A tends to Z.
to 2.3.4 and 2.2.4,
Call o i the transposition
this group is amenable.
of the points
i and i+l.
The group B is generated by the set of all o i where if we denote by T the unit translation, between
the following relations
•
=
°
Tl°°0°T-i
G is generated by the two elements
extension
o 0 and T. It is amenable
of an amenable group by an amenable
It remains
to be proven that the kernel
F is finite there exists
Let k be an integer The mapping homomorphism Then,
as an
(2.3.3).
group.
an integer n such that h(T n) = e F.
and K be the finite part
from S K to B given by
group
of every group h o m o m o r p h i s m h
from G in a finite group F is not a solvable Because
hold
these generators:
qi Thus,
i is in Z; and,
{0,...,k-l}
o --> ~.Tnk.~.T -nk
and its image is contained
for every k, the kernel Ker(h)
in the kernel
contains
.
is a one-to-one of h.
a subgroup
isomorphic
to
S K and is therefore not solvable.
2.4. REFERENCES
For the existence and Krylov
of invariant probability measures,
to Bogoliubov
(I).
Invariant means on groups are examined by Greenleaf Markov-Kakutani also by Bourbaki groups).
refer
theorem is discussed (3)
(I).
of course by its two authors but
(with a generalization
to the case of solvable
34
Regarding paper
the existence
of the ameaning filter,
(I) or refer to Moulin Ollagnier
see F~Iner's
and Pinchon
original
(2,7) for an alter-
nate proof and a study of locally compact amenable groups.
3. ERGODIC THEOREMS
3.1. INVARIANT LINEAR FUNCTIONALS
3.1.1. Ergodic theorem. Let E be a real vector space, s a sublinear function, and T a right action of an amenable group G by linear one-toone mappings preserving s on E. Then, for every f of E, the following holds: Sup ~(f) ~eK(E,s,G)
=
Inf AeF(G)
(I/IAI).S(gYAf.Tg)
=
lim sup (i/IAI).s( [ f.T g) M geA
Proof. Since G is amenable, there exist linear invariant functionals bounded by s on E. Denote by K(E,s,G) the convex and compact set consisting of all these functionals and let ~ be one of them. Then, for every f in E, ~(f)
=
(I/IAI).~( ~ f.T g) geA
and the two inequalities easily follow Sup ~(f) ~eK(E,s,G)
~
Inf AeF(G)
(I/nAl).s( X f.T g) geA
lim sup (I/IAI).s( ~ f.T g) M geA Consider the function p on E given by p(f)
=
lim sup (i/IAI).s( X f'Tg) M geA
So defined, p is a sublinear function and, according to 1.1.7, is the least upper bound of all linear functionals below it.
36
In order to complete
the proof,
convex compact sets K(E,s,G) The first inclusion,
we have to show the equality of the two
and K(E,p).
K(E,s,G) ~ K(E,p),
tivity property of s gives
is already proven.
the inclusion of K(E,p)
The subaddi-
in K(E,s).
Now the only thing to be proven is that a linear functional, p on E, is invariant,
i.e. equal
to 0 on the elements
bounded by
f-f.T h with f in E
and h in G. What is true for f is also true for -f and we have only to show that p(f-f.T h) is equal
to 0.
Let us calculate p(f-f.T h)
=
lim sup (I/IAl).s( M
~ (f-f.Th).T g) geA
lim sup (i/IAl).(s(f)+s(-f)).m{h}(A) M 3.1.2.
Corollary.
As the upper
limit along the ameaning
the theorem above is equal to the greatest tity on the set of all non-empty
filter used in
lower bound of the same quan-
finite parts of G, this upper
limit is
in fact a limit. 3.1.3.
Example.
continuous
Consider
the usual vector
near function
s defined
for every continuous
a right action that preserves
function
to the limit p(f) of the ergodic
3.1.4.
Corollary.
number p(f), For an upper
Proof.
on X, is equal
along M
I foT ). geA g
In the above situation,
functions
it is possible
functions
on X with values
semi-continuous
measure ~ on X, ~(f) eventually
averages
not only for continuous
semi-continuous
s on C(X).
f on X, the least upper bound of all
~(f), where ~ is an invariant Radon probability measure
(I/IAl)'s(
of all
space X, and the subli-
in 1.1.3. When acting on X by homeomorphisms,
the amenable group G induces Then,
space C(X) consisting
real functions on a compact Hausdorff
to define the
on X, but also for upper
in the interval
E-~, +~]
function f and an invariant Radon probability
is well defined by regularity;
and p(f), which is
equal to -~, is still the least upper bound of all ~(f).
Consider
the upper
the value ~ (f) to ~.
semi-continuous
function on K(C(X),s,G)
giving
37 Because
function p is non-decreasing,
corollary
1.3.7 of Dini's
lemma
gives us Sup ~eK(C(X),s,G) The converse 3.1.5.
inequality
Definition.
~(f) holds
A real
=
Inf ~eC(X),~f
and the proof
function
p(~)
~
p(f)
is achieved.
c on the set F(G) of all finite parts
of G is said to be subadditive
if c(~)
decomposition
of a finite part A as a combination
indicators following
of the indicator
of subsets inequality c(A)
Function
F(G)
Remark.
coefficients,
of
1 A = ~ ~BIB,
the
~ ~BC(B)
The subadditive there
Remark.
to 0 and if, for every
holds:
g of G, the numbers
for which
3.1.7.
of it with positive
c is said to be invariant
every element 3.1.6.
g
is equal
if, for every finite part A of G, and c(A)
functions
is a sublinear
According
and c(Ag)
are equal.
are exactly
extension
to lemma 2.2.15,
the functions
on
to the cone C~(G).
strongly
subadditive
functions
are subadditive. 3.1.8.
Lemma.
Let B be a finite non-empty
part A of G, the following I{ geG,BgcA} I The positive
<
IAI
function A B on F(G) &B(A)
=
The set
=<
hold:
I{ geG,Bg0A#@} ]
is then defined by
I{geG,BgnA#@,Bg~A}I
and the limit of the ratio Proof.
inequalities
part of G. For every finite
{geG,BgaA}
(I/IAI).AB(A) is exactly
along M is equal the intersection
where b is an element
of B; and the number
is less than or equal
to the cardinal
The set
{geG,Bg~A~}
cardinal
number
of A.
of all b-IA,
of the elements
number
to 0.
of this set
of A.
is the union of all b-IA where b is in B; and the
of this set is of course
greater
than or equal
to the one
38
On the other hand, A B is bounded by a sum of standard elements AB(A)
~
~ l{geb-IA,~b'eB beB b~B
Proposition.
g~b'-iA}l
mBb-l(A)
and the limit along M of the ratio 3.1.9.
of Z
(I/IAI).AB(A)
is therefore
0.
Let G be an amenable group and c a subadditive
invar-
iant function on F(G). Then,
the greatest
not empty,
lower bound of all averages
(I/IAI).c(A),
is equal to the limit of the same averages
where A is
along the ameaning
filter. Proof.
It is quite clear that the greatest
equal
to the upper limit along the ameaning
We must merely prove,
for every non-empty
(I/IBI).c(B) Consider
therefore
positive
combination IA
Dividing
(I/IBI).
properties
[ c(B) Bg~A
(I/IAI).I{g,BgcA}I
(I/IAI).I{g,BgOA#~, Remark.
continuous
Bg~A} I
+
allow us to write Sup c(C).[{geG,Bg~A#@,Bg~A} C=B
I
tends to i along M and the ratio
tends to 0 along M according
The use of the ergodic
function on a Hausdorff
alternate proof of the previous It is indeed possible
theorem 3.1.4 with an upper semi-
proposition. subadditive
function
space X, an action T of G on X by homeomor-
and an upper semi-continuous
finite part A of G,
to lemma 3.I.8.
compact set would have led to an
to find, for every invariant
on F(G), a compact Hausdorff phisms
I A as a
of subsets of B:
by IAI and taking the limit along M lead to the result because
the first ratio
3.1.10.
~
of the indicator
of right translates
l i Bg~A#@ Bg~A
and invariance
c(A)
finite part B, the inequality
decomposition
of indicators
(i/IBl).
filter.
lim sup (I/IAI).c(A) M
the following
=
The subadditivity
~
lower bound is less than or
function f on X such that, for every
39
c(A) Consider
=
s( [ foT g) geA
the compact Hausdorff product
action of G on Y by translations
space Y = [-~ ,c({e})]G,
and the closed invariant
the
subspace X,
defined by all the inequalities
~e n(g) g A The triple
<
c(A)
(X,T,f), where T is the restriction
and f is the coordinate
function
looked for in order to complete 3.1.I1.
Remark.
all M(D,~),
The ameaning
to X of the action of G
at the unit element e of G, is what we the proof.
filter,
that we defined with the basis of
can also be defined by other basis.
For every finite part
ml,...,m n
number ~, M(m I ..... mn;e)
denotes
of the cone E, and every positive the non-empty
real
subset of F(G) consisting
of all finite non-empty parts of G for which all differences
(l
are
less than
(I/]AI).mi(A) These
sets constitute
convergence
result
- q(mi)
a basis
of the ameaning
filter according
to the
3.1.9.
For every finite non-empty part B, function n B is given by
nB(A) So defined, F(G)
=
n B is an element of E; and the sets M(nB;e) , for all B in
and all positive
Indeed,
the following
nB(A) 3.1.12.
Remark.
arbitrarily
IB.AI
real numbers
E, constitute
a basis of M.
inequality holds with any given element b 0 of B:
-]AI
__< [BI.mB.boI(A)
Let B be a finite non-empty
invariant with A, which means
part of G. Then,
that,
B.A is
for every element M I of
M, there exists another element M 2 of M such that B.A belongs whenever
A belongs
to M I
to M 2.
Given a finite part B of G, the intersection invariant with A. In the first case, use the M(nD;c)
~ b-IA
is also arbitrarily
beB as a basis of M and observe
that
40 nD(B.A ) The following
=
nD.B(A)
inequality
I
<
then gives
(i/I B.AI ) .nD(B.A)
On the other hand,
=<
(i/l AI ) .nD.B(A)
the inequality
mD( ~ b-iA) beB which
the proof of the first result:
-I(A) mB'D'bo
< =
is true for any given element
b 0 of B, leads
quickly
to the proof
of the second result. 3.1.13.
Example.
functions
i
g G (2.1.3) with Denote
Given an element m of Z, it is possible
on the compact
Hausdorff
the increments
therefore
space T(G)
by g$ the past of g for the order 3, i.e.
in G that are strictly
The greatest
lower bound of all increments iA(T) a
=
strongly Denote
to G\{a},
the set of
-
m(a~A)
of G that do not contain
also the limit of these
a is, when A
increments
because m is
subadditive.
by ia(T)
functions
this limit;
on T (depending
as a greatest
Because
of the property
bounded
below.
lower bound of continuous
solely on the restriction
finite part of G), i a is an upper
semi-continuous
of capacities
(2.2.10
of the order
function
(iv)),
If T is the action of G on T(G) by translations, property
on
less than g in the order 3.
m({a}U(a~A))
on the set of all finite parts tends
some
of m.
all elements
simply
to define
of all total orders
to a
on T(G).
this function
the following
is
invariance
holds: ig = ieoTg
One can easily verify s( [ i ) geA b When G is an amenable
that =
m(A)
group,
the ergodic
theorem shows
that the least
41
upper bound of the values by all Radon invariant But this particular lity measure
Indeed,
probability
example
~ on T(G),
~(i e ) because
taken on the upper
=
measures
is richer;
the following
semi-continuous is equal
function
e
to q(m).
for every Radon invariant equality
i
probabi-
actually holds:
q(m)
i e is the directed
greatest
lower bound of the continuous
functions
i A and because every Radon measure is regular there exists e ' ' for every positive real number ~, a finite part A 0 such that u(i~-i e) is less
than ¢ when A contains
Then,
A 0.
for every finite part A, re(A)
Integrating
=
I geA
over T(G)
(m((g$0A)U{g})
-
for the probability
m(g$~A)
)
~ gives -I
geA
geA
We arrive at the following
(I/IAI).m(A)
inequality:
=< e
+
~(i e)
+
(I/IA]).mA0(A).~(ie~-i e)
Take the limit on A along M, and let e tend to 0 to derive q(m) As the converse the equality
3.2.
~(ie)
inequality
follows
from the ergodic
VECTORS
Description.
AND MEAN ERGODIC
In this section,
we consider
a normed vector
space
group G on E by continuous
linear mappings.
We are therefore MA(f)
the proof of
THEOREMS
(E,II If) and a right action T of an amenable one-to-one
theorem
is achieved.
INVARIANT
3.2.1.
~
interested
of a given element
in the behaviour
f of E along
of the ergodic
the ameaning
filter
averages
42
MA(f) In the general
=
(I/IAI).
case,
~ f.T g geA
the closed hull C(f)
generated by f and its images
f.T g is not compact. 3.2.2.
Proposition.
suppose
We consider
the previous
that the norms of the operators
situation
and, moreover,
T g are uniformly bounded
(e.g.
they are isometries). Then,
there exists at most one invariant element
invariant hull C(f) Proof.
of an element
By definition
f of E.
the closed convex invariant hull C(f)
of the convex set of all convex combinations non-negative
in the closed convex
is the closure
~ ~(g) f.T g, where
~ is a
real function with a finite support and an integral
to I for the counting measure Every invariant
continuous
linear functional
If i I and i 2 are two invariant vectors linear
on E is constant on C(f).
of C(f),
for every continuous
invariant
In order to conclude
that il-i 2 is actually
have to prove that,
functional
~(il-i 2) is equal
continuous
to 0
~ on E.
the null vector of E, we
for every invariant vector of E different
there exists an invariant
equal
on G.
linear functional
from O,
~ on E such that
~(i) ~ O. The uniform bound property means
that there exists a positive number K
such that the norms of all T g are bounded by K. We can then construct respect
an invariant norm N on E which is continuous
with
to the initial norm N(f)
Applying
=
Sup Ilf.Tgll ~ geG
K. Ilfll
theorem 3.1.1 produces Sup ~eK(E,N,G)
~(f)
Using these equalities p(i) According functional extension
=
=
p(f)
=
Inf AeF(G)
(I/IAI).N(
for the invariant element
N(i)
=
to the Hahn-Banach given on Ri by
[ f.T g) geA
i, the following holds:
Ilill > 0 theorem,
~(i)=p(i)
~ is a linear functional
it is possible
to extend the linear
to the whole space E so that the bounded by p; the linear functional
43
is invariant and continuous 3.2.3.
Proposition.
and, moreover,
and ~(i)
Taking the hypotheses
assuming
Proof.
of the previous
the existence of an invariant
closed convex invariant hull C(f) norm-limit
is strictly positive.
of f, this single element would be the
We have to state =
0
Because 0 is the only invariant vector of C(f-i)
every continuous
riant linear functional
on E is equal to 0 on f-i.
Restricting
to the elements
this result
3.2.4.
=
Sup ~(f-i) ~eK(E,N,G)
since the initial norm Von Neumann's
theorem.
action of an amenable
=
ameaning
filter.
lira (I/IAI).N( [ (f.T g- i)) M geA
Let H be a Hilbert
space and T a right
group G on H by isometries. averages MA(X)
The limit is the orthogonal
closed subspace H G consisting
Proof.
and appling theorem
II II is bounded by N, the proof is achieved.
For every vector x of H the ergodic
under
of K(E,N,G)
inva-
we get 0
Then,
in the
along M of the averages MA(f).
lim (I/IAI).II I (f.Tg - i)~ M geA
3.1.1,
proposition
element
of all vectors
converge
projection
along the
of x on the
of H, that are invariant
the action of G. The family of norms of all average operators
by I; and the set of all x, for which the averages the orthogonal
projection
of x on HG,
Let us first show the convergence If x belongs
to HG,
If x has the form a computation orthogonal
the property
converge along M to
is then a closed subspace
result for special
of H.
vectors of H.
clearly holds.
y-y.T h, the averages
that we just did several
projection
is uniformly bounded
converge times.
to 0 along M thanks
The null vector
to
is the
on H G of such an x because y-y.T g is orthogonal
every invariant vector. It remains
to be proven that the subspace of H generated by H G and the
increments
y-y.T g is dense in H, and therefore
subspace reduces
to 0.
that its orthogonal
to
44
Let then z be a vector orthogonal Because
z is orthogonal
It is then orthogonal 3.2.5.
Mean ergodic
to H G and to all y-y.T g.
to the increments
it is an invariant vector.
to itself and therefore equal to 0.
theorems
in L p. Let
(X,&,~,T)
be an abstract dynamical
system where T is an action of an amenable group G; let ~ be the o-algebra of all invariant
events
in ~ and p be a real number greater
than or equal
to I. Then,
for every f in LP(x,~,~),
the ergodic averages
norm along the ameaning filter; tion
E(fl~)
of f with respect
The arguments theorem.
in the L pexpecta-
to the o-algebra ~.
of the proof are very similar to those of the Von Neumann's
The conditional
expectation
is a projection
with a norm i and the average operators I. The subspace
of LP(x,~,u)
this subspace
contains
ments
where h belongs
y-yoTh,
converge
and the limit is the conditional
elements
in LP(x,a,~) to
for which the result holds is then closed;
the invariant elements of LP(x,&,~)
and the incre-
to G and y to LP(x,~,~).
In order to prove the norm-density by the invariant
operator
have a norm less than or equal
in LP(x,a,~)
and the increments,
of the subspace generated
we have to examine
two
possibilities. First,
suppose that p is strictly greater
than i and consider
the conju-
gate number p' of p (I/p + I/p' = I). An element of LP'(x,
of L p', orthogonal
Let ~ be such an element. (l~I)(P'/P).sgn(~)
when p is equal
because
the measure
3.2.6. tems,
equal
result.
functions
norm-convergence
Because ~ is orthogonal
to I, an invariant
is finite;
to the invariant
element # of L ~ belongs
it is then orthogonal
The norm-convergence
For instance,
it is possible
continuous
element element to L I
to itself and
to 0.
Counterexample.
general
is an invariant
of L p, ~ is equal to 0.
Second,
therefore
to all the y-yoTh,
,~).
of averages
in the case of topological
to construct
counterexamples
are constant
and in which, however,
of the averages
is in no way a dynamical
sys-
in which all invariant there is no
of a given function along M to a constant
function. The next theorems make this remark more explicit. 3.2.7.
Theorem
(Furstenberg).
Let
(X,T) be a topological
dynamical
system,
45
where T is an action of an amenable group G on the compact Hausdorff space X by homeomorphisms. The two following properties
are equivalent:
I)
there exists a single invariant
2)
for every continuous to a constant
Proof.
Radon probability measure
function f, the ergodic averages
function.
It is clear that the second of these two properties
first because
implies
the
the value taken by the probability measure on the function
is the same as the value taken on the constant. to the constant According
on X
converge
itself and therefore
to the Hahn-Banach
p(f)
+
p(-f)
clearly determined.
theorem,
Radon probability measure yields, =
1~is value is then equal
the uniqueness
of the invariant
for every f in C(X), the equality
0
that is to say
limM xexSUp (i/IAI).geA ~ f°Tg This is equivalent to a constant 3.2.8. ergodic
limM xexinf (i/IAl)'g~Af°Tge
to the uniform convergence
function;
Definition.
=
and hence property
A topological
if the equivalent
dynamical
properties
of the averages
I implies Froperty
along M 2.
system is said to be uniquely
of theorem 3.2.7 hold for it.
3.2.9. Definition.
A topological
gically
if there exists a point x in X whose orbit under the
transitive
dynamical
system is said to be topolo-
action T is dense in X. This property clearly implies
that all invariant
continuous
functions
are constant. 3.2.10.
Example:
a topologically
ergodie topological
dynamical
transitive,
but nonetheless
Let G be an infinite amenable group. The dynamical 2.1.2 is not uniquely ergodic because different
probability
lity measures.
vectors
not uniquely
system. all Bernoulli
lead to different
system described
in
schemes built on
invariant
Radon probabi-
46 Let us first show that the only continuous constants,
invariant
and then solve the counterexample
For simplicity's two elements
sake, consider
functions
that we mentioned
are the earlier.
the case where the alphabet I has only
and denote them by +I and -I.
Call a x the coordinate
function at the point x of G given by Ox(~)
and denote by OA the product of the coordinate
functions
= $(x)
at all points
of the finite part A. The family of all functions OA is an orthonormal space L2(X,~,u),
where the abstract
scheme built on the probability When f is a continuous
dynamical
vector
basis of the Hilbert
system is the Bernoulli
(1/2,1/2).
function the following
inequality holds:
2 I (ffoo A d~ ) AeF(G) Moreover,
if f is invariant,
~
ff2 du
the equality
= holds for every g of G, and because of the preceding
inequality,
scalar product is equal to 0 for every non-empty
the
finite part A
of G. The following
equality
According
=
then holds for every A in F(G): .
to the Stone-Weierstrass
theorem,
the °A generate
space of C(X); and the two linear functionals are the same.
a dense sub-
on C(X), and .~,
Then /f2 d~ and the continuous
=
(ff d~ )2
function f is almost everywhere
equal to a constant;
and because ~ gives a positive mass to all open sets, f is actually a constant
function.
The same arguments constants;
show that the only invariant
and the abstract
dynamical
When the group G is countable,
system
elements
(X,~,~,T)
in L 2 are the
is ergodic.
the compact Hausdorff product space X has
a countable basis of open sets consisting
of the cylinder
sets [A,A],
47
where
[A,A]
is the closed and open subset of X [A,A]
=
{$eX,~xeA E(x)=+I,VxeA\A
The set D of all points
~(x)=-l}
in X whose orbit is dense is then the countable
intersection D
=
~ [A,A]
=
{ ~ ,3geG Tg (~) O[A,A]#~}
~
( U
[A,A]
(rg)-i ( [A'~ ) )
geG
All invariant events of the previous
intersection
measure which is then equal to i because cal system. Thus, ~(D) is equal to I, which implies topological
dynamical
have a strictly positive
of the ergodicity
of the dynami-
that D is not empty;
system is then topologically
3.3. INDIVIDUAL
ERGODIC THEOREMS
In the previous
section we were interested
and the
transitive.
in the asymptotic
behaviour
of the averages (l/n). where
(X,Q,~,T)
[ foT i 0
is an abstract dynamical
system with an action of Z and
where f an element of LI(x,~,u). In particular, conditional
we showed the norm-convergence
expectation
iant events. For this very special known,
Birkhoff's
of f with respect to the o-algebra
sequence of averages
theorem,
according
for every dynamical
another convergence
~ foT i k
=
n
result is also holds
to the mean ergodic theorem.
system with an action of Z, the norm-conver-
gence still holds for the sequence of averages (l/n).
to the
of all invar-
to which the convergence
almost everywhere. But this result is far from being analogous Indeed,
of these averages
48
where
(kn) is an arbitrary
The norm-convergence
sequence of integers.
is derived from the right invariance
(M(D,6))
of the ameaning
However,
it is possible
to shift the Birkhoff's
averages by a sequence
(k n) in such a way to lose the almost everywhere To do this,
Emerson
irrationnal
rotation of the one-dimensional
The counterexample
of the basis
filter.
(2) considers
the dynamical
convergence
result.
system generated by an
torus R/Z.
that we give next has still another nature
deals with generalized
Bernoulli
3.3.1.
Let ~ be the product
Counterexample.
since it
schemes. space R Z endowed with the
product o-algebra ~ of all Borel o-algebras. Given an element
i of Z, denote by X i the random variable
to the coordinate
at the point i.
If ~ is a probability measure on
(~,00
a sequence
translate
(An ) of disjoint
are integrable
the law ~ symmetrical and their expectation
Let T be the unit shift. convergence
=
with a first moment is equal
so that the X i
to 0.
theorem in L I gives
the norm
(i/n). ~ X i ieA n
Because
the A n are disjoint, According
the random variables
to the Borel-Cantelli
a the probability
that the upper
lemma,
M n are mutually
indep-
for a given positive
limit of the M n is greater
real than
to a is equal to I if and only if the series of the probabilities
of the events
(Mn~a)
If this holds
for every a, the upper
where
finite parts of Z such that A n is a
The mean ergodic
endent.
almost
P
of the averages M n to 0 w h e n n tends to infinity Mn
or equal
a single probability
independent with the same law ~.
of the segment {0,..,n-l}.
Choose moreover
number
on R, there exists
such that the X i are mutually
Consider
corresponding
diverges.
surely infinite
limit of the sequence M n is then
and this sequence
cannot converge almost every-
to 0.
To complete
the proof of the counterexample,
cal probability
law on R with a first moment
real number a, the following ~n
series diverges:
we must construct such that,
a symmetri-
for every positive
49
(u
*n
denotes
the n-th convolution
In the following propostion, 3.3.2.
Proposition.
power of u).
we construct
this law.
For every real number ~ strictly between
the function exp(-[t[ ~) is the Fourier
1 and 2,
transform of a probability
law
on R. For the corresponding
probability u~,
the function
(Ix[) 8 is integrable
if the real number 8 is less than ~. Moreover,
this law is a stable one,
independent
random variables
homothetic
n((l-~)/~).X
This probability Proof. Because
the law of the average
with a law ~
of a variable
gives the example
it is continuous
The positive
is the same as the law of the
that we looked for in 3.3.1.
and takes the value according
type is a classical
proven in several ways. of section XXII.I. the Fourier
from the properties
of n
X with a law p~.
The function t --> exp(-[t[ ~) is of a positive
transform of a probability,
Because
i.e.
It is possible,
of the "El6ments
1 at 0, it is the Fourier
to Bochner~s
result
theorem.
in Probability for instance,
d'Analyse"
transform is one-to-one,
type.
Theory that can be to solve exercise
by Dieudonn4.
the stability of p~ comes
of the function exp(-]t]~).
The law of the average
of n independent
random variables with a law pe
is then that of the homothetic (n ((I-~)/~) ) .X
of a random variable whose
law is ~ .
If we denote by ~ the Fourier the Fubini's
transform of p~, a simple application
theorem shows that the integral (I
- Re~(t))/t
of
of
B+I
is the product of the integral
over R for the Lebesgue measure
of
(I - cos u)/u B+I
by the expectation
for p~ on R of Ix[ B
If the real number ~ strictly for the Lebesgue measure
lies between 0 and 2, the integral
of the function
over R
2
50
(i - cos u)/u B+I
is finite and the two other integrals It is then easy to see that is strictly The series
are then simultaneously
Ixl B is integrable
for ~
finite
if and only if B
less than ~. that we are interested
as the integral
for ~
in has,
for every a, the same nature
of the function
Ixl (=-II~) and therefore
diverges.
The probability ~ 3.3.3.
Remark.
amenable tion.
group,
then provides
The existence
us with the counterexample
of ameaning
for which Birkhoff's
Only partial
results,
sequences
sought.
of finite parts of an
result holds,
is still an open ques-
such as for groups with a polynomial
growth,
are known.
3.4. THE SADDLE ERGODIC THEOREM
This section is devoted pological 3.4.1.
dynamical
to the proof of a minimax
sytems; we give this result in a very general
Saddle ergodic
theorem.
subset of a real locally convex Hausdorff
following
to toform.
Let K be a convex and compact non-empty
T be a left action of an amenable one mappings
theorem related
topological
group G by affine
on K, and let f be a real function on
vector
space;
continuous F(G) x K
let
one-towith the
properties:
i)
for every x in K, the partial
function
in the sense given in definition ii)
for every finite part A of G, the partial is concave and upper semi-continuous
iii)
f(.,x)
this function
is invariant,
=
function
f(A,.)
on K
i.e. for every g in G, every
finite part A of G and every point x of K, f(Ag-l,Tg(x))
is subadditive
3.1.5
f(A,x)
51
Then,
the following
Inf AeF(G)\{~}
minimax
(iii AI ) .Sup
f(A,x)
for the subset
action
of G.
Proof.
For every element inequality f(A,x)
of all invariant
of K under
the
Sup f(A,t) teK
of these subadditive
the limits
points
x in K G and every finite part A of G, the
along M of these averages taking
(ill AI ) .f(A,x)
Sup Inf xeK G AeF(G)\{~}
holds: ~
Also the averages Because
=
xeK
where K G stands
following
equality holds:
invariant
functions
and the limits
are in the same order.
along M are equal
to the greatest
lower bounds,
the least upper bound on all x in KG, we get the inequality
in
one direction Sup xeK G
Inf (I/IAl).f(A,x) AeF(G)\{ ~}
In order
to prove
function
on F(G)
the converse defined
T(A)
=
decomposition
lower bound,
=
call ~ the subadditive
and also the limit along M, of the
is nothing
to prove.
fix a finite part B of G and consider
of the indicator
iA
inequality,
by
ratio (I/IA I).~(A). If ~ is equal to -~, there case,
Inf (I/IAI).Sup f(A,t) AeF(G)\{ ~} teK
Sup f(A,t) teK
Let = be the greatest
In any other
=<
(I/IBI).
the positive
of A:
~ IBgQA BgNA#~
For every x in K, the subadditivity
property
of f(.,x)
leads
to the
inequality f(A,x) Setting
apart
and using
~
(I/IBI).
~ f(Bg~A,x) Bg~A#~
in the sum the elements
the invariance
property
g for which Bg is contained
of f, we get
in A
52
f(A,x)
~
(I/IBI).(
X f(B,Tg(x)) Bg~A
+
% f(BNAg-I, Tg(x)) BggA
Choose an x such that f(A,x) = T(A) and call x A the center of gravity of all Tg(x) where Bg is contained in A. Because the partial function f(B,.) is concave, the following holds: (I/I AI ) .]~(A)
=<
(i/I BI ) .f(B,XA) • (i/I AI ) • I{g,BgcA} I
+
(i/I AI ). (I/I B1 ) .AB(A) .Sup ]~(C) C~B
Let y be a limit point of the x A along M. Use the upper semi-continuity ot f(B,.) and lemma 3.1.8 to derive (I/IBI).f(B,Y)
~
Moreover, according to remark 3.1.12, the intersection O(b-IA,beB) becomes arbitrarily invariant with A; and y is an invariant point. The following inequality then holds for every finite non-empty part B: Sup (I/IBl).f(B,y) yeK G
~
We conclude the proof, according to corollary 1.3.7, by replacing the ratio (i/IBl).f(B,y) with its greatest lower bound on the set of all finite non-empty parts of G.
3.5 REFERENCES
About general ergodic theorems, refer to Aloaglu and Birkhoff (I). Furstenberg (I) gives the theorem on uniquely ergodic dynamical systems. Almost every ergodic theory text gives a proof of Birkhoff's theorem; refer, for instance, to Billingsley (i). This theorem was generalized, especially to groups with a polynomial growth, by Chhtard (I), Bewley (i) and Emerson (2). The frame of the proof of the saddle ergocic theorem given here was inspired by Misiurewicz (I), and we shall have the opportunity to speak about it further in Chapter 5.
4. ENTROPY OF ABSTRACT
4.1.
EQUIVALENCE
OF ABSTRACT
DYNAMICAL
We shall classify here the objects abstract
dynamical
of an equivalence 4.1.1.
system between
Definition.
and U are actions
(X,~,~,T)
DYNAMICAL
SYSTEMS
given in the definition by giving several
two dynamical
Two dynamical
SYSTEMS
systems.
systems
(X,~,~,T)
and
(Y,~,~,U),
of the same group G, are said to be isomorphic
exist an element X' of the o-algebra ~ whose measure of the a-algebra ~ whose measure mapping
= (i.e. measurable
and Y',
such that a) X' is invariant under
the action T of G
b) Y' is invariant under
the action U of G
for every g in G,
mapping ~oT g
bimeasurable
instead of the provisory
in 2.1.4
(where X' = X and Y' = Y).
Definition.
Suppose that
distance
notion of an iso-
one of a strict isomorphism
Let ~ be a Boole algebra,
and d the corresponding
i.e.
ugo~
From now on, we shall be concerned with this particular morphism,
between X'
~ commutes with the actions, =
where T if there
is i, an element Y'
is I, and a one-to-one
together with its inverse mapping)
c) the bimeasurable
4.1.2.
2.1.5 of an
different notions
P an additive
introduced
function on ~,
on ~ (d(A,B) = P(AAB)).
(~,d) is a complete metric
of a group G on ~ by one-to-one mappings
space,
and consider
that preserve
an action T
the distance
and
the Boolean operations. This triple 4.1.3.
(~,d,T)
Example.
is said to be a dynamical
If (X,~,~,T)
is a dynamical
Boole algebra.
system,
the quotient
algebra
of the o-algebra ~ by the o-ideal Afof all events whose probability
is
54
equal
to 0, together with the distance
metric
space,
d(A,B)
= ~(AAB),
and the given action of G immediately
is a complete
induces
an action on
this Boole algebra a/~ by automorphisms. 4.1.4.
Definition.
An isomorphism between
associated with two dynamical
systems
the dynamical
Boole algebras
is called a conjugacy of the two
systems. 4.1.5.
Example.
It is evident
that an isomorphism
in the sense of defi-
nition 4.1.1 induces a conjugacy of the systems. 4.1.6.
Remark.
In questions
often only interested A condition systems
sufficient
involving dynamical
in the associated
isomorphic
unit interval
these spaces are isomorphic
together with the Lebesgue measure;
hence
We shall suppose this condition to be fulfilled whenever interpretation encounter
of the results;
and all explicit
are actually Lebesgue
The metric
ones,
together with Borel probabilities.
~/~ has no atom,
[0,I]
Boole algebras.
is that the spaces are Lebesgue
to Polish spaces
algebra
dynamical
we are in fact
to deduce a conjugacy between two dynamical
from an isomorphism
quotient
systems,
systems
i.e.
If the to the their name.
we need a point that we shall
spaces.
space a/~ corresponding
to a Lebesgue
space is separable.
4.2. ENTROPY OF PARTITIONS
4.2.1.
Definition:
The function
entropy of probability
defined on the interval ] 0 ~
extended
to the whole segment
negative
function n is strictly concave;
the value 0 at the points Function n is subadditive, numbers whose
<
If all the coordinates vector
by
n(x)
in a continuous
= -x.Log(x) function.
in particular,
can be
This non-
it only takes
0 and I. i.e. for every pair
sum is less than or equal
n (x+y)
nical base,
F0 ~
vectors.
n (x)
+
(x,y) of positive
real
to I,
n (Y)
of a vector of R n are non-negative
and if the sum of all these coordinates
is said to be a probability
vector
in the cano-
is equal
of order n.
to i, this
55
It is always vector
possible
of greater
Consider
to consider
order by setting
~
l~i~n
are added
of the properies
its value
vectors
(indepen-
is the same when null
with a given order;
strong
subadditivity
i, j and k range
coordi-
and continuous
on
it is also symmetri-
the probability
abc, and consider
property
natural
numbers
vector
and suppose
p = (Pijk) whose
also the probability
vectors
that the
order
is the product
q of order a, r of order
as follows:
q = (pi)
where
r = (Pij)
where Pij =
~ Pijk
s = (Pik)
where Pik =
~ Pijk J
Then we get the inequality
of entropy.
from I to a, I to b and I to c.
ab and s of order ac, that are defined
Pi = j!kPijk
H(p) + H(q)
~ H(r) + H(s)
We have to state
i,
,k
inequality
n(Pijk )
+
is equivalent
~n(p i) l
=<
.[ n(Pij) l,j
to the following
i,~,kPijk.Log(Pij.Pik/Pijk.Pi) where
in R n is
of n, function H is concave vectors
Let a, b and c be three positive
This
to 0.
whose
concave.
Proposition:
Consider
vectors
to a vector.
cal and strictly
Proof.
coordinates
on the set of probability
because
all sets of probability
subscripts
vectors
as a probability
n(pi)
Function H is well defined dently of their order)
4.2.2.
all supplementary
on the set of probability
H(p) =
Because
vector
the function H on the set of all probability
definition
nates
a probability
the sum only ranges
not equal
over the triples
to 0; this results
the logarithm (Pij.Pik/Pi)
because is equal
~
the sum over all triples to i.
! n(Pik) i k
one:
0
(i,j,k)
from the concavity
+
for which Pijk is
and the monotonicity (i,j,k)
of the ratios
of
56
The assumption that no Pi is equal to 0 is in fact not a restriction. 4.2.3. Definition. A finite partition of the probability space (X, ,~) is a finite set {PI' .... Pn } of events such that the measure of the intersection of Pi and Pj has a measure 0 when i and j are different, and such that the measure of the union of all Pi is equal to I. 4.2.4. Definition. A partition Q is said to be finer than a partition P if every atom of Q is contained (modulo 0) in an atom of P. This property is denoted by
Q >> P.
4.2.5. Remark. The previous relation ">>" is a preorder relation on the set of all finite partitions of a given probability space; and the equivalence classes of the corresponding equivalence relation (P>>Q and Q>>P) can be identified with the partitions of the unity between non-null idempotents in the quotient algebra ~/~. We shall identify a partition with its equivalence class for the previous equivalence relation. 4.2.6. Proposition.
For the previous order relation, every two parti-
tions have a least upper bound. Proof. The partition R, which is defined by its atoms
Rij = Pi~Qi, is
clearly a least upper bound for the set (P,Q). 4.2.7. Definition. The entropy of the partition P = (Pi) is the entropy of the probability vector (~(Pi)); this entropy is denoted by H(P). 4.2.8. Proposition. Entropy is a strictly increasing function on the ordered set of all partitions of a given probability space. Proof. Suppose Q finer than P and use the subadditivity of n to derive H(Q)
=
~ ( ~ n(~(Qj))) l Qj Pi
=> ~ n(~(Pi)) l
Equality can only hold if, for every atom Pi of P, the number n(~(Pi)) is the sum of all n(~(Qj)) where Qj is contained in Pi" Strict concavity of the logarithm then implies that this last equality is only possible when one of the ~(Qj) is equal to ~(Pi); and entropy is then strictly increasing.
57 4.2.9. Proposition. Entropy is strongly subadditive on the set of all finite partitions, i.e. verifies, for every triple (P,Q,R) H(PfQVR)
+
H(P)
__< H (PVQ)
+
H (PVR)
Proof. This proof is apparent from the strong subadditivity property of H on the set of all probability vectors (proposition 4.2.2). 4.2.10. Definition.
Given a o-algebra
6 contained in ~ a n d
a finite par-
tition P of (X,~,~), the conditional expectations E(IPiI~) constitute a partition of the unity almost everywhere. The integral of the entropy of this random probability vector is called the conditional entropy of P with respect to D and is denoted by H(PI6): H(PI~)
=
4.2.11. Proposition.
[ I n(E(ip.I~)) l
If the o-algebra~
d~
is generated by a finite parti-
tion Q, we denote by H(PIQ) the conditional entropy H(PI~Q). The following equality then results from an immediate calculation: H(PIQ)
=
H(PVQ)
H(Q)
4.2.12. Remark. The entropy H(P) can be considered as the conditional entropy with respect to the trivial o-algebra {~,X}. 4.2.13. Proposition. of the o-algebra,
Conditional
entropy is a non-increasing
i.e., if ~ contains ~,
H(P]~) ~ H(P]~).
Proof. According to definition 4.2.9, one has H(P]~)
=
[ / n(E(ip. IZ)) d~ l
=
I ] E(n(E(Ip. I~))113) d~ l
And because n is a concave function H(PI~ )
_<
~ f n(E(E(!Pil~) 16)) d~ l=
function
58
The composition property of conditional expectations gives, for every subscript i, the equality
E(E(IPil~) I~)
=
E(iPiI~)
The summation in the right-hand side of the above inequality is then equal to the conditional entropy H(PI~). 4.2.13. Theorem. Let (6n) be an increasing sequence of o-algebras contained in ~, and let 6 be the c-algebra generated by all ~n; for every finite measurable partition P of (X,~,~) the convergence result holds: H(P[6)
=
Inf (H(P[~n))
=
lim H(P[~ n) n --> =o
A proof can be found, for instance, in the work of Parry (2).
We shall need in the following some control over the concavity of the entropy. The next lemma provides us with it. 4.2.14.~ Lemma. Let (~i) be a convex combination and, for every i, let (p~)~ be a probability vector. Denote by (pj) the mean value of all these probability vectors
pj
=
i ~ ~iPj
Then the linearity defect
H(p)
~iH(P l )
is bounded by the entropy
of the probability vector (=i). i Proof. Consider the probability vector ($ij = ~iPo )" The mean value of the entropies of the p l is equal to the conditional entropy of P with respect to ~, i.e. the difference between the entropy of Q and the entropy of ~; hence the result.
59
4.3. ENTROPY OF DYNAMICAL
4.3.1.
Definition:
Consider
SYSTEMS
mean entropy of a partition.
a dynamical
system
(X,~,~,T)
where
(X,~,~)
is a probability
space
and T an action of an amenable group G on this space by automorphisms. Given a finite partition P of the space and a finite part A of the group, pA stands for the least upper bound of the partitions ments of A, the symbol PoT g representing
PoT g for all ele-
the partition whose atoms are
the inverse images of the atoms of P by the automorphism The equality of the probability comes from the invariance these two partitions The following
associated
of the measure ~ under T; and the entropies
+
is the strong subadditivity H(p A~B)
=< H(P A)
The real function on F(G), A --> H(pA), strongly
subadditive
T g.
to pA and to pAa of
are then equal.
inequality
H(p AUB)
vectors
+
of entropy
H(P B)
is then positive,
and left invariant;
non-decreasing,
this function belongs
to the
cone Z. By definition
the mean entropy of the partition
P in the dynamical
tem (X,~,~,T)
is the mean value of the previous
invariant
the greatest
lower bound of the ratio
sys-
capacity,
i.e.
(I/IAI).H(P A) on the set of all
finite non-empty parts of G and also the limit along the ameaning
filter M
of the same ratio. We denote the mean entopy of the partition P by h(P). 4.3.2.
Definition:
entropy of dynamical
The entropy of a dynamical
entropies h(P) of all finite partitions 4.3.3.
Remark.
By its very definition,
if two dynamical
4.3.4.
Remark.
systems.
system is the least upper bound of the mean
systems
of the measure
entropy is a conjugacy
are conjugate,
their entropies
The mean entropy h(P) of a partition
but the entropy of a dynamical
Sinai theorem shows that the entropy of a dynamical partition.
We have first to define the notion of a generator.
invariant:
are equal.
is always finite,
system may be infinite.
the mean entropy of a generating
space.
The Kolmogorov-
system is equal to
60 4.3.5.
Definition.
A finite measurable
ting for the dynamical
system
given by d(A,B) = u(AAB), translates
partition P is said to be genera-
(X,~,~,T)
if, for the pseudo-distance
the Boole algebra generated by P and all its
PoT g is dense in ~.
When G is countable,
a dynamical
system with a generator
is a Lebesgue
space. 4.3.6.
Proposition.
partitions
(P,Q) is a distance; Proof.
The function defined on the set ~ of all finite
of the space =
(X,~,~) by
H(PIQ)
+
H(QIP)
we call it the entropic distance.
The symmetry property of ~ is apparent.
is the triangle inequality;
we use,
of the entropy and its monotonicity H(P[ R)
Thus,
=
H(PVR)
<
H (PVQVR)
<
H(PVQ)
=
H ( P IQ)
function ~ is a distance,
therefore,
Definition.
is a measurable
H(R) -
H(R) H(Q)
+
+
4.3.8.
Definition.
H(QVR)
-
H(R)
H(QIR)
providing,
An ordered partition
mapping
point
the strong subadditivity
to calculate
we are concerned here with the equivalence 4.3.7.
The only remarkable
of course,
that we specify that
classes defined in 4.2.5.
of the probability
space
(X,O,p)
from X to a finite set I.
Given two measurable
partitions
p and q of (X,~,p)
which are ordered by the same finite set I, we denote by d(p,q)
the
number (1/2). ~ ~(p-l(i)Aq-l(i)) ieI So defined,
d is a distance on the set of all partitions
ordered by I.
61 4.3.9.
Lemma.
Let p and q be two measurable
same set I with n elements; Then the following
~ (P,Q) Here ~ stands on
[0,I]
Function
d between
distance
of P
p and q:
non-decreasing
~(x) = sup(n(t),0~t~x)
~ is concave
partitions.
the entropic
2n2.~ ((2/n2) .d(p,q) )
for the non-negative
given by
and it is constant Proof.
holds between
distance
=<
ordered by the
call P and Q the associated
inequality
and Q and the previous
partitions
on the segment
continuous
real function
.
[0,~,
it agrees with n on [ 0 , I / ~
on [i/e,l].
We look for an upper bound of the difference
H(PFQ)
- H(Q),
which
can be written H(PVQ) This expression
- H(Q)
=
~(n(~(Pj~Qj))-n(~(Qj))) j
can be bounded,
n.~(d(p,q)/n)
+
thanks
+
to the concavity
[ n(~(Pi~Qj)) i#j of ~, by
n(n-l).~(d(p,q)/n(n-l))
and then by n2.~(2d(p,q)/n 2) The same is true for the difference the demonstration
of the previously
H(PVQ)
- H(P);
all of which
stated relation
between
leads to
the numbers
6
and d. 4.3.10.
Proposition.
P is a generator
for the dynamical
if and only if the set of all partitions is dense in ~ for the entropic
coarser
system
(X,&,~,T)
than some upper bound pA
distance.
Proof. I) Let Q = (Qi' lJi
Q~ of ~, that are unions
of elements
of pA,
!
every i, ~(QiAQi ) is less than ~. Define
a partition
QJ
=
Q", with n+l elements,
QJ \
j Qk'
if
l<j
by setting
such that,
for
82
Q~+I Partition
=
x \
Q" is coarser
the comparison
than pA. Thanks
of distances,
real number ~, another than 60,
~ Q'.' l~jSn j
the entropic
to the previous
it is possible
positive distance
real number between
to find,
lemma concerning
for every positive
~0 such that, when ~ is less
the given partition
Q and Q" is
less than e. 2) Let QI be an event
in ~ and Q2 its complementary
part.
Let Q be the
partition whose atoms are Q1 and Q2 and let Q' be a partition than pA whose entropic distance to Q is less than ~. The greatest tends
lower bound of all distances
coarser
from Q1 to the atoms of Q'
to 0 with ~.
4.3.11.
Remark.
The mean entropy h(P)
is clearly
a non-decreasing
function
of P. 4.3.12.
Proposition.
For every finite non-empty
part B of g, the mean
entropy h(P B) is the same as that of P. Proof.
The ratio
(IB.AI/IAI)
the set B.A becomes
tends
arbitrarily
to 1 along
invariant
the ameaning
with A (remark
filter
3.1.12);
and hence
the result. 4.3.13.
Proposition.
is a Lipschitz
With respect
to the entropic
function with a constant
distance,
mean entropy
1 on ~.
Proof. For every finite part A of G, the repeated H((PVQ) A) Hence
-
H(p A)
~
use of subadditivity
leads
AI .(H(PVQ)-H(P))
the two inequalities h(PVQ)
=< h(P) + H(QIP)
And then the inequality h(P)
- h(Q)
;
h(P~Q)
=< h(Q)
+ H(PIQ)
that we seek ~
h(PVQ)
- Inf(h(P),h(Q))
<
Sup (H(Q] P) ,H(P] Q) )
<
6 (p,Q)
to
63
4.3.14.
Kolmogorov-Sinal
the dynamical
system
theorem.
(X,~,~,G),
If the partition P is a generator
for
the entropy of the system is equal to
the mean entropy of P. Proof.
The mean entropy is a continuous
distance,
and the subpartitions
function on
for the entropic
of the pA are dense in
; the entropy of
the system is then the least upper bound of the mean entropies particular
of these
partitions.
On the other hand,
the entropy is non-decreasing
and we need only take
the least upper bound on the set of all pA, which all have a mean entropy equal
to h(P).
4.3.15. 2.1.8,
Example.
To resume the example
the partition
open sets
of Bernoulli
({$(e) = i},iel)
is a generator
To prove this result,
notice
topology and remember
the regularity
for every Bernoulli
that the cylinder
scheme.
Radon measures.
to the mean entropy of the
at the unit element.
The mutual of H(pA);
in
sets form a basis for the
of the invariant
The entropy of such a system is then equal partition
schemes described
at the unit element of G whose atoms are the closed-
independence
of all translates
and the entropy of the abstract
of P leads to the additivity dynamical
system is then the
entropy of the probability
vector on which the Bernoulli
4.3.16.
stated the theory of entropy as an invariant
Remark.
for dynamical termine were
Kolmogorov
systems.
Until
if the two Bernoulli
scheme is built.
this time there had been no argument schemes built on (1/2,1/2)
and
to de-
(1/3,1/3,1/3)
isomorphic.
We will recall
that the spectral
noulli
in the case of the group Z (cf. Billingsley
schemes
4.3.17.
Remark.
ral question Bernoulli Ornstein
properties
After this first success
arose:
schemes
Is entropy a complete
are the same for all Ber(I)).
of the entropy theory, invariant
a natu-
in the class of all
?
solved this problem
in the case of Z: Two Bernoulli
schemes
with the same entropy are isomorphic. The essential
arguments
Shannon-McMillan Several
of the proof are the mean ergodic
theorem,
and the Rokhlin
authors have been interested
the action of an arbitrary amenable space in order to develop We presented
the
lemma.
in generalizing
these results
to
group on a standard probability
an isomorphism
the mean ergodic
theorem,
theorem.
theorem in chapter
3 and shall
speak about
64
Rokhlin's devoted
lemma in chapter 8. The end of this chapter, however,
to the Shannon-McMillan
4.4. THE ALMOST SUBADDITIVE
Definition.
theorem.
ERGODIC THEOREM
AND THE SHANNON-McMILLAN
4.4.1.
THEOREM
Given a probability
able partition P of it, the information positive
space
(X,~,~)
relative
and a finite measur-
to P is the measurable
function I(P) defined by I(P)
=
~ (-Log(n (p)) .i peP P
This function is integrable the partition 4.4.2.
will be
and its expectation
Theorem.
of an infinite of this space.
Let (X,~,~,T)
be a dynamical
system where T is an action
amenable group G. Let P be a finite measurable
The mean information
(I/IAI).I(P A) converges
filter and the limit is an invariant The limit is also the greatest of the averages all invariant
is the entropy H(P) of
P.
in Ll-norm along the ameaning
function.
lower bound of the conditional
(I/IBI).E(I(pB)I~)
partition
expectations
with respect to the o-algebra
~ of
events.
The convergence
result is the Shannon-McMillan
theorem that was proved
for the group Z. The general proof of this result for an amenable group was first given by Kieffer
(I). The characterization
lower bound and the demonstration work
(8) and essentially
4.4.3.
Definition.
elements
decompositions
be a dynamical
system.
ergodic theorem.
A family
(fA) of
indexed by F(G), is said to be almost subadditive
a positive real number C, such that, of every indicator
IA = the difference
depend on the almost subadditive
Let (X,~,~,T)
of LI(x,~,~),
if there exists
of the greatest
that we give here follow on earlier
for all positive
of a finite part A, such as
I ~BIB Bel
(fA - Bel~~ BfB) is almost negative,
i.e. the measure
65
u((fA-
[ ~BfB )+) Bel
of its positive part is less than or equal to C. 4.4.4. Definition. A family (fA) of the previous type is said to be covariant if, for every finite part A of G and every element g in G, f(Ag) = f(A) oT g
4.4.5. Theorem. Let (X,&,~,T) be a dynamical system and P a partition of the measure space. Then, the family of elements of LI(x,~,~) defined by fA = I(pA) is almost subadditive in the sense of definition 4.4.3; and the constant C can be taken equal to I. Proof. Consider a decomposition with positive coefficients:
IA
=
~ ~BIB BeF
The difference l(p A) -
~ ~BI(P B) BeF
is constant on every atom of pA and can be written (~(B(a)) ~B) I(pA) -
~ ~B I(PB) BeF
=
~ A L°g(BeF aeP
).I a
~(a)
Here B(a) stands for the atom of pB that contains a. When a real number x is greater than or equal to I, it is an upper bound for its logarithm. We can then bound the positive part of the previous difference in the following way: (~(B(a)) xB) (l(pA) _
~ ~BI(pB))+ BeF
< =
~ (BeF aePA+
). i ~(a)
a
Here the subscript a varies only in the set P+A of all atoms of pA for which the logarithm is greater than or equal to 0.
66 Adding some positive terms, we get v((I(P A) -
~ ~BI(pB)) +) BeF
<
~ A ( ~ (v(B(a)) aeP BeF
~B
))
where a now varies in the whole set of all atoms of pA. It now remains to prove that the number H (~ (B(a)) XB) ) aeP A (BeF is bounded above by 1 for every positive decomposition of a finite part A. We shall use an induction argument to prove it. The result is obviously true when A reduces to a single point; but when this is not the case, distinguish a point g in A and denote by A' the complementary part of {g} in A. Putting together the atoms of pA contained in the same atom of pA', we can write the previous sum this way:
a
~pA( H (~(B(a))XB)) = ~ A,( ~ (~(B(a')XB).~A ~ ( ~ ((B(a))~B))]) BeF a'e~ B A' '(a)=a ' geB
The sum of all coefficients ~B of the parts B containing g is equal to I; according to H~Ider's inequality every bracket CA
~ ( H ((B(a)) ~B) )] '(a)=a' geB
is bounded above by
geB
( ~ (~(B(a))) ~B) A'(a)=a'
Given an atom a' of pA' P the union of the B(a) for all atoms a such that A'(a) = a' is exactly B'(a'), where B' is the part B A' of A'. The initial expression (~(B(a)) ~B)
a~pA(BeF is then bounded above by a,~pA,(e B'eF' g (u(B'(a'))
XB'
)
67 where the B' = A'~B verify
IA' = B'~F '~B'IB' This last expression is less than or equal to i, by induction hypothesis. The proof of the almost subadditivity of the family (I(pA)) of the information functions is now complete. 4.4.6. The almost subadditive ergodic theorem. Let (X,~,~,T) be a dynamical system where the group G is amenable and infinite. Let (fA) be an almost subadditive and covariant family of elements of LI(x,~,~)
indexed
by the finite parts of G, and let (fA) verify the greatest lower bound property: the greatest lower bound K of all (I/IB[).~(f B) is finite Then the average (I/IAl).f A converges in Ll-norm along the ameaning filter. The limit is an invariant function and it is the greatest lower bound of the conditional expectations (I/IAI).E(fAI ~) with respect to the o-algebra ~ of all invariant events. A brief remark before proving the theorem - the convergence result is still true (it even becomes trivial) when the group is finite, but examples can easily be found where the limit is no longer the greatest lower bound of the conditional expectations. Proof of the theorem. The proof shall be divided in three steps. a)
Let us first show that the averages asymptotically become less than
the conditional expectations; fix a finite part B and consider the difference A(A,B) A (A,B)
=
(I/IAl).f A
(i/I BI).E(fBI~)
As a function of A, the positive part of A(A,B) tends to 0 along the ameaning filter. Indeed, consider the usual positive decomposition 1A
=
(I/IBI).
~ IBg~A Bg~A@~
68 Apply tile almost subadditivity
inequality
to the corresponding
elements
of the family to derive ~((fA - (I/IBI)" Setting apart the elements the covariance
~ f~ ^A) +) Bg~A#~ ~ g ~
< =
C
g such that Bg is contained
in A and using
property of the family, we get the inequality
~(A(A,B) +)
~
C/IA[
+
II(I/[BI).E(fB[~)
+
(I/IA[).(I/]B I) •Bg~A#~,Bg,A [IfB Ag -I °TglIl
- (I/IAI).(I/IB[).B~cAfBoTg]II
As we shall see, the three terms of the right side tend to 0 along the ameaning
filter.
The first,
the ratio C/IAI,
amenable
group G is infinite;
as regards
tends to 0 because
the second,
the mean ergodic theorem to (i/IBl).f B demonstrates to 0 along M; and as for the third, because
the application
of
that this term tends
there are only finitely many
different parts of B, and because ~ is invariant, bound ~(B)
the
there exists a common
for the norms
II(fB Ag-l)oTgIll and as the third term is bounded by (I/IA I). (I/IB I).~(B).AB(A) it also tends to 0 along M. b)
Because L 1 is a complete
the functions
space,
in order to prove the convergence
(I/IAI).f A along M, we need only verify Cauchy's
of
criterium.
Let then E be a positive real number and B be a finite part of G such that
(I/IBI).~(fB)
According
is less than
K + e/6.
to the first step of the proof,
it is possible
part D of G and a positive real number ~ such that than e/6
whenever A belongs
to M(D,~).
to find a finite
~(&(A,B) +)
is less
69 We then get an upper bound for the integral (I/IBI).u(E(fBI~))
of (I/IBI).E(fBI~)
=
(I/IBI).U(fB)
<
K + e/6 (I/IAI).~(fA)
Hence a lower bound for the integral -e/6 The integral
<
~(A(A,B))
=
of A(A,B)
~(A(A,B) +) - ~(A(A,B)-)
of the negative part of A(A,B)
and the Ll-norm of A(A,B)
+ E/6
is then less than c/3
is less than e/2.
Then, when the two finite parts A' and A" belong to M(D,6), II(I/IA'I).fA,and Cauchy's c)
(I/IA"I).fA,,II 1
condition is fulfilled.
From the right invariance
the limit f is invariant.
of the M(D,~),
the conditional
a contraction
expectations
(I/IAI).E(fAI~).
along M
expectations
on
parts of G).
Proof of theorem 4.4.2.
To complete hypotheses Indeed,
this proof, we notice that the family of the almost subadditive
this family is almost subadditive
the greatest
(I(pA)) verifies
all
ergodic theorem.
It is covariant because ~ is invariant
according
to theorem 4.4.5.
under the action of G.
lower bound property holds since all these functions
are non-negative. 4.4.8.
is
(i/IAI).f A is
(I/IAI).E(fAI~)
lower bound of the same conditional
the set of all finite non-empty
Last,
expectations
operator of L I, the limit f of the averages
(and the greatest
deduce that
expectation with repect to a given o-algebra
also the limit of the conditional
4.4.7.
we immediately
Using the first step of the proof, we see that
f is less than or equal to all conditional Because
<
Remark.
probability
We have only defined finite mesurable
partitions
of a
space.
The notion of a countable measurable
partition
is also important.
70 Such a partition is said to have a finite entropy if the series of all numbers n(~(p))
converges.
By definition,
the sum of the series is then
the entropy of the countable partition. The Shannon-McMillan theorem holds even for countable partitions with a finite entropy and there is therefore no change in the proof. 4.4.9. Remark.
For particular
sequences of finite parts,
countable amenable groups, and for special
it is possible to state Breiman's theorem thus:
The convergence of the mean information relative to a given partition not only holds in norm but also almost everywhere. Breiman's work concerned the usual means on Z. For groups with a polynomial growth,
Ornstein and Weiss,
and Derriennic all obtained significant
results.
4.5. REFERENCES
As regards conjugacy and isomorphism of dynamical lingsley
systems, refer to Bil-
(I), who gives the proofs of several properties of entropy.
On this subject,
see also the works of Kolmogorov
the article by Rokhlin The martingale
(i), of Sinai
(i), and
(2).
theorem for the conditional entropy is proven by Parry (2).
The proof of the isomorphism of Bernoulli schemes with the same entropy was acccomplished by Ornstein
(1,2). Refer also to Smorodinsky
The first generalization of the Shannon and McMillan theorem case of amenable groups is found in Kieffer
(i).
(i) to the
(i); the current author
(8)
has given an alternate proof of this extension and the characterization of the limit function as a greatest lower bound of conditional expectations.
5. ENTROPY AS A FUNCTION AND THE VARIATIONAL
5.1. TOPOLOGICAL
5.1.1.
PRINCIPLE
ENTROPY
Definitions.
Let (X,T) be a topological
dynamical
system.
Given
a finite open cover 0 of X, N(O) stands for the logarithm of the minimal number of open sets in all subcovers
of O.
For every finite part A of G, denote by 0 A the open cover of X by all sets (Tg)-l(ag) geA where the a
are the elements of O. g The set function A --> N(O A) is weakly subadditive,
i.e. whenever
A
and B are disjoint parts of G, N(O AUB)
~
N(O A)
+
N(O B)
The mean topological
entropy h(0) of the cover 0 is the upper
along M of the ratio
(I/IAI).N(oA);
The topological bound h(T)
entropy of the dynamical
(eventually
limit
and h(0) is finite. system is then the least upper
equal to +~) of the mean topological
entropies
of
all finite open covers of X. 5.1.2.
Remark.
morphism dynamical
Topological
of topological
entropy is an invariant number for the iso-
dynamical
metric entropy of abstract a topological
5.1.3.
two isomorphic
systems have the same topological
Let us state a result analogous
parating
systems:
dynamical
open cover,
Definition.
The topological
entropy.
to the Kolmogorov-Sinal
dynamical
topological
systems:
theorem for the
the topological
system is the mean topological
entropy of
entropy of a se-
if one exists.
The variational
principle
entropy of a topological
the least upper bound of the entropies
gives the following dynamical
system
of all abstract
result:
(X,T) is
dynamical
72
systems
(X,~,~,T) built with invariant Radon p r o b a b i l i t y measures.
We know that the topological In the present chapter, principle
we shall state a more general variational
for the pressure of a continuous
5.2. PRESSURE OF A CONTINUOUS
5.2.1.
entropy is the pressure of the null function.
however,
Definitions.
function.
FUNCTION AND THE V A R I A T I O N A L PRINCIPLE
Let X be a compact Hausdorff space and T be an action
of an amenable group G by h o m e o m o r p h i s m s
on X.
Denote by W the set of all open symmetrical neighborhoods A of XxX;
of the diagonal
set W is a basis of the u n i f o r m filter.
Let 6 be an element of W. A subset E of X is said to be ~-separated if any two distinct elements of E are not neighbors
of order ~.
An open cover O is said to be of order ~ if the Cartesian square of every element a of 0 is contained in ~. Because X is compact,
there exist finite open covers of X that are of
order ~. Any m a p p i n g i from a ~-separated set E into a finite open cover O of order 8, such that the image of every point of E is one of the open sets of 0 to which this point belongs,
is one-to-one.
Such mappings do exist and a 6 - s e p a r a t e d set is then finite. Let f b e a real function on X w h i c h is bounded above. For every finite subset E of X, Z(f,E)
Z(f,E)
=
stands for the sum
[ exp(f(x)) xeE
For every finite open cover 0 of X, Z(f,O)
Z(f,O)
=
stands for the sum
[ Sup exp(f(x)) aeO xea
W h e n e v e r E is a 6-separated set and 0 an open cover of order 6, there exists a one-to-one mapping from E to O, leading to the inequality Z(f,E)
~
Z(f,O)
73 The least upper bound Pl(f,~) set of all f-separated
of all numbers Z(f,E), where E runs in the
subsets of X, is finite; and the greatest lower
bound P2(f,~) of all numbers Z(f,O), where 0 runs in the set of all finite open covers of order ~, is not equal to 0. Thus the following inequality clearly holds: Pl(f,~)
$
P2(f,f)
If A is a finite part of G and f an element of W, {A stands for the following element of W: fA
=
~ (TgxTg)-I(~) geA
Here T is the initial action of G on X and TxT the component-wise
action
of G on the square XxX. If A is a finite part of G, fA is the real function on X given by fA
=
[ f°Tg geA
Denote then by pl(f) the upper limit of the ratio along the ameaning fiter. If ~' is finer than ~, the f-separated
(I/IAI).Log(PI(fA,fA))
sets are also 6'-separated
and we
get the inequality Pl(f,6)
__< Pl(f,6')
That allows us to define the function Pl on the set of all real functions that are bounded above on X by pl(f)
=
Sup pl(f,~)
=
~eW
Similarly,
p2(f,f)
lim pl(f,~) W
is the upper limit of the ratio
along M. Here also, when ~' is finer than ~, we get p2(f,~)
<
p2(f,~ ')
and P2 is defined by p2(f)
=
Sup p2(f,f) feW
=
lim p2(f,~) W
(I/[AI).Log(P2(fA,6A))
74 For every 8 in W, pl(f,6) where
~
p2(f,8)
$
exp(s(f)).h(O)
0 is any open cover of order 6.
Then for every real function pl(f) where h(T)
~
p2(f)
~
is the topological
5.2.2.
Theorem.
system
(X,T) be finite
numbers
pl(f)
Proof.
Because
is bounded
entropy
of the dynamical
entropy
system
(X,T).
function
dynamical
on X. The two
are then equal.
f is continuous,
it is uniformly
exists,
continuous
for every positive
on the compact
real number
~, an
6 of W such that - f(y)J
<
c
whenever the pair (x,y) belongs to 8. Choose an element y of W such that the composition in 8. In order
to calculate
mal y-separated
the upper bound Pl(f,y),
subsets
square yoy
simply
of X; they are y-spanning,
consider
i.e.
is contained the maxi-
the y-neighbor-
of all points make an open cover of X.
Such a cover
is of order 8; hence P2(f,~)
Applying, whose
we get
of the topological
and let f be a continuous
therefore
If(x)
hoods
above,
exp(s(f)).h(T)
Let the topological
and p2(f)
space X. There element
f on X which
<
the inequality
exp(c ) .PI (f,Y)
for every finite part A of G, this result
composition
square yAOYA
P2(fA,SA) Take the logarithm limit along
<
<
in 8A, we get
exp([A[ .E).PI(fA,YA)
of both members,
the ameaning p2(f,8)
is contained
to 8 A and chosing YA'
filter pl(f,y)
divide
them by I A[ , and take the
to get the double +
c <
pl(f)
+ c
inequality
75 Take now the limit along W to derive
p2(f) Because p2(f)
<
pl(f)
this inequality
is true for every e, the two numbers
5.2.3.
Definition.
5.2.4.
Remark.
a continous
The pressure
5.2.5.
The equivalence
function
and
Function
t gives
Deduce
is a Lipschitz
tool in studying
the
t (we can no longer
call
of the increments
~
on X, p(f+h)
is
PI(fA,~A).Sup exp( [ hoTg(x)) xeX geA
function
as a function
p(f + hoT g - h)
entropy p(0)
on C(X):
=
is finite,
f on X and the pressure
for every h in C(X)
p(f)
functions
of the pressure:
from the finite rank inequality
function with constant
is infinite,
of
of the fil-
p(f) + t(h).
If the topological
it verifies,
If p(0)
control
to the sum
nite for every continuous wing properties
function
(f,h) of real continuous
the relation
Corollary.
of the pressure
property
3) on C(X) given by
Pl((f+h)A,~ A)
5.2.6.
f is the
lim (I/IA I ).Sup ~ hoTg(x) M xeX geA
less than or equal
Proof.
is a decisive
the sublinear
the following
For every pair
of the ameaning
this property
Consider
=
function
and p2(f).
on the space C(X).
it p as we did in chapter t(h)
pl(f)
of the two definitions
is independent
as a function
Theorem.
of the continuous
of the two numbers
ter M. On the contrary, pressure
on X.
pl(f)
are equal.
common value p(f)
and,
+
function
p(f)
is fi-
has the follo-
p is non-decreasing;
i with respect
it
to the uniform norm;
and every g in G,
p(f)
is also infinite
for every continuous
function
76 The control
of the increments
responsible
for all these properties.
5.2.7. where
Variational principle.
of the function p by the function t is
Let
(X,T) be a topological
dynamical
system
the space X is compact and where T is an action of an amenable
group
on X by homeoemorphisms. The pressure p(f)
of a continuous
bound of all sums
(h(~) + ~(f)),
dynamical
to the Riesz representation
The proof is divided
in two parts:
h(~) + ~(f)
5.2.8.
<
inequality
Theorem
gical dynamical
theorem.
theorem to be given in 5.2.8;
shall be a consequence
(Variational
principle
and the
of theorem 5.2.13.
- part one). Let
(X,T) be a topolo-
system and ~ be an invariant Radon probability measure. is measurable
given by the Riesz representation
theorem,
itive real number e, there exists a symmetrical the diagonal
with respect
open neighborhood
+ ~(f)
~
p2(f,~)
on the use of the quantity R(~,~) which gives a measurement
for the overlap ratio of an open cover with respect Let us study this number before proving Definition.
a one-to-one mapping j from P into ~ such that
every atom Pi of P is contained Definition.
in the corresponding
open set j(Pi ).
Given a probability measure ~ and a finite open cover
~, the overlap ratio R(~,~) entropies
to a given probabi-
the theorem.
A finite partition P is said to be adapted to the open
cover 6 if there exists
5.2.10.
of the
of a finite open cover of X. The proof of this theorem depends
essentially
5.2.9.
~ of
+ E
Proof. We shall look for 6 as the union of the Cartesian squares
lity.
to the
and for every pos-
such that
h(~,P)
elements
~ on
the first inequality
For every finite partition P of X which a-algebra
is the entropy of the abstract
p(f)
from the approximation
converse
where h(~)
system built with the invariant Radon probability measure
X, according
results
real function on X is the least upper
is the least upper bound of all conditional
H(p,PIQ) where P and Q are measurable
partitions
adapted
to 5.
77
5.2.11.
Proposition.
cover, which means (whose elements
The overlap ratio is a subadditive
function of the
that the overlap ratio R(~,~IV6 2) of the cover ~IV~2
are the non-empty
intersections
alna2,
with a I in 6 1 and
a 2 in 6 2 ) is less than or equal to the sum of the overlap ratios R(~,~ I) and R(~,~2). Proof.
Let P be a partition adapted
a one-to-one mapping j = (jl,J2) (considered Gluing
subpartition one,
of the covers
as the sets of their open sets).
together
Similarly,
to the cover 61V6 2. There then exists
from P into the product
the atoms of P which have the same image by Jl' we get a
P' which is
adapted
we get a subpartition
to 6
I" P" adapted
to 6 2 . Because j is one-to-
P is the upper bound of P' and P".
A similar
construction
can be made for the second partition Q.
Using the subadditivity respect
H(~,PIQ) Because
property
to a given a-algebra,
=< H(~,P'IQ)
the conditional
a-algebra,
expectation with
+ H(~,P"IQ)
expectation
is also a decreasing
=< H(~,P'IQ')
By taking the least upper bounds of the subadditivity
+ H(~,P"IQ")
in both members we complete
Proof of theorem 5.2.8. Let
Let P be a finite partition
(X,T) be a topological
Because ~ is outer regular,
with respect
to the
theorem. of the atoms of P
these open sets can be chosen in such a way
than a given positive
cover adapted
dynamical
on X.
of the difference between an atom and its neighborhood real number.
The partition P, with which we started, is adapted
is measurable
an open cover by taking open neighborhoods
that the measure is smaller
of X which
given by the Riesz representation
We construct
the proof
of R(~,6).
system and ~ be an invariant Radon probability measure a-algebra
function of the
we derive H(~,PIQ)
5.2.12.
of the conditional
we get
is by the construction
of the
to it and, for every finite part A of G, the partition pA
to the cover ~A"
Let 5' be an open cover of order 6A, let R be a partition
adapted to ~'
i.e. finer than the cover 6A; and
78 Consider a mapping 0 from 6' into 6 A that sends every open set of ~' to one of the elements of 6 A that contain this open set. If j is one of the admissible one-to-one mappings from R into 6', consider the subpartition Q of R obtained by the following construction: an atom of Q is the union of all the atoms of R
whose image by 0oj is
equal to a given element of ~A" Partition Q is adapted to 6 A and we then get H(~,P A)
<
H(~,Q) + H(u,PAIQ)
Hence H(~,P A) + ~(fA )
<
H(~,Q) + R(~,6 A) + u(fA )
<
H(~,R) + R(~,6 A) + ~(fA )
According to the Jensen inequality, than or equal to Log(Z(fA,6')) H(~,P A) + ~ (fA)
the sum (H(~,R) + ~(fA )) is less
and we next obtain
=< R(~,~ A) + Log(P2(fA,6A))
Using the subadditivity of R and taking the upper limit along M, we derive h(~,P)
+
~(f)
<
R(~,6)
+
p2(f,6)
If the least upper bound of the measures of all differences between the atoms Pi of P and their open neighborhoods is less than (I/n).Sup(~(Pi)), every partition which is adapted to the cover has exactly n elements. Thanks to lemma 4.3.9 the entropic distance ~(P,Q) between two partitions, both adapted to the cover 6, is bounded above by 2n2.~(2d(p,q)/n2). Hence an upper bound for the overlap ratio R(~,6)
<
2n2.~ (2 Sup (~ (Oi\Pi)) )
The previous relation is true whenever the right-hand side is less than (I/n).Inf(~(Pi)). The proof of the first part of the variational principle is thereby complete.
7g 5.2.13.
Theorem
(Variational
open neighborhood
~ of the diagonal,
probability measure
- part two).
there exists an invariant
>
greatest
Radon
pl(f,~)
Here we use what are essentially
the saddle ergodic
For every symmetrical
~ on X such that
h(~) + ~(f)
Proof.
principle
theorem.
lower bounds
the arguments
We have to be careful,
for the proof of
however,
because
some
are no longer reached.
Let us then fix a strictly positive
real number a.
For every finite part A of G it is possible
to find a ~A-separated
set
E such that
Z(fA,E)
~
PI(fA,~A).exp(-a)
Call a A the probability measure
on X whose
support
is E and which gives
to a point y of E the mass
OA({y})
=
(I/Z(fA,E)).exp(fA(y))
Denote by ~A the average of the translates
~A
=
(I/]AJ).
of °A by the elements
of A
~ Tg(o A) geA
Then let N be a filter with the following properties: first,
N is finer than the ameaning
second,
the ratio
filter M;
(I/IAJ).Log(PI(fA,~A))
converges
to pl(f,~)
along N; and, This
third,
the probability measure
~A has a weak limit along N.
limit ~ is an invariant probability measure,
as we have just seen
(proof of theorem 3.4.1). In order to state that this limit verifies consider
a finite measurable
to the boundaries
the anticipated
inequality,
partition P such that ~ gives a measure
0
of the atoms of P.
For every atom Pi of the partition P, the real function v --> ~(Pi ), which is defined on the set of all Radon probability measures continuous
at the point ~.
on X, is
80 Moreover,
we can construct P of order ~ so that every atom of pA contains
at most one point of E, which leads to H(OA,P A) + ~A(fA)
=
Log(Z(fA,E))
~
Log(PI(fA,~A))
- a
Entropy is a strongly subadditive set function; it is subadditive, according to remark 3.1.7 and we can then exploit the usual positive decomposition
IA Subadditivity
=
(I/IBI).Bg~#¢IBg~A
leads to the following inequality
(I/[AI)'H(eA ,PA)
~
(I/IAI)" I (I/IB])'H(eA ,PBg) geA
+
(I/IAI).Log(n).(IB.A
\ {g,BgcA} I)
where n is the number of the atoms of the partition P. As a function of the probability, (I/IAI).Log(PI(fA,6A))
entropy is concave; hence ~
(I/IBI).H(~A ,PB)
+
~A(f) + (I/IAl).a
+
(I/IAI).Log~n).(IB.A
Taking the limit along N, and using the continuity of
\ {g,BgcA}l)
~ --> H(~,P B) at
the point ~, we obtain pl(f,~)
<
(I/IBI).H(~,P B) + ~(f)
Now, taking the least upper bound on the set of all finite parts B, we get the following inequality: pl(f,~)
<
h(~,P) + ~(f)
and the second part of the variational 5.2.14.
Remark. We assumed,
<
h(~) + ~(f)
principle in proven.
in the previous proof,
the existence of
finite measurable partitions of the space X which were of order ~, such that the measures for ~ of the boundaries of the atoms were O.
81
To demonstrate clearly that such partitions do exist,
take an open cover
finer than the given one. Now we still must show that it is always possible to find a m e a s u r a b l e set between a compact set and a given open neighborhood of it, with a null boundary for the measure ~. Therefore,
consider a continuous
function ~ on X whose values
lie between
0 and i, which is equal to 1 on the compact and to 0 out of the open neighborhood.
There are only countably many real numbers
1 such that the ~-measure of the set
{xeX,~(x)
If u does not belong to this exceptional
= t}
t b e t w e e n 0 and
is not equal to 0.
set, the b o u n d a r y of the measur-
able part {~ > u} is contained in the null set {~ = u} and we achieve the result we sought.
5.2.15. Example.
Consider the symbolic system
right translations,
X = IG
where G acts by
and let Y be a closed invariant subset of X.
The p a r t i t i o n at the unit point of G is also a separating open cover. The pressure of a continuous
p(f)
=
lim sup M
function f is then equal to
(I/IAI).Log( [ A Sup exp(fA(~)) aeI SA=a
For a finite part A of G, p(fA,PA) (~(fA)+H(~,PA))
)
is the least upper bound of the sum
on the set of all invariant Radon p r o b a b i l i t y measures,
if P is the p a r t i t i o n at the unit point. Employing the saddle ergodic theorem 3.4.1 gives the variational principle in this particular
5.2.16.
situation.
Remark. When we encountered an open cover ~ as a neighborhood of
the diagonal of XxX, we implicitly supposed that an open cover 5' order ~ is finer than 6 in the usual
sense,
of
i.e. every element of 5' is
contained in an element of ~. This
is m e r e l y a set-theoretic result which can be settled by an easy
induction argument:
If
A, A I ..... A n
AxA of the product
are some parts of a set S such that the part SxS is contained in the union of the AixAi,
there exists an i such that A is contained in A.. i
82
5.3. ENTROPY AS A FUNCTION OF THE MEASURE
5.3.1.
Definition.
We are now interested
in some properties
of the func-
tion ~ --> h(~), which is defined on the compact convex set M(X,T) isting of all invariant Radon probability measures space X. We suppose that the topological dynamical 5.3.2.
system is finite,
Proposition.
entropy of the topological
thus the previous
If the topological
finite and if the group
is infinite,
cons-
on a compact Hausdorff
function is bounded.
entropy of the dynamical
system is
the entropy is an affine function of
the measure. Proof. We must simply state the following equality (~i,~2)
of invariant Radon probability measures
of strictly positive representation
for the a - a l g e b r a
+ ~2.h(~2,P)
=
given by the Riesz
h(p,P)
such a partition P is also measurable
from ~i and ~2 because respect
(=i,~2)
to I, and every
theorem
~l.h(~l,P) Indeed,
on X, every pair
real numbers with a sum equal
finite partition P that is measurable
for every pair
these measures
for the a-algebras
are absolutely
continuous
built with
to ~.
The least upper bound on the set of all these partitions limit and therefore Calculate
the equality
then the difference, (I/IAI).(H(~,PA)
This positive
difference
to lemma 4.2.14,
still holds
is a directed
for the limits.
for a finite part A of G,
_ = I . H ( ~ I , P A ) _ =2.H(~2,PA))
is bounded above by (i/IAl).Log(2),
and tends to 0 when the cardinal
according
number of A tends to
infinity. 5.3.3.
Definitions.
Given an element 6 of W, a partition P is said to be
strongly of order ~ if the Cartesian of P is contained
square of the closure of every atom
in a.
The mean entropy of order 6 of an invariant Radon probability measure (or, better,
of the dynamical
bound h(~,6)
of the mean entropies h(~,P)
which are strongly of order ~.
system built on ~)
is the greatest
of all measurable
lower
partitions
83
5.3.4. Lemma.
Given an invariant Radon p r o b a b i l i t y measure ~ and an ele-
m e n t ~ of W, h(~,6)
is the greatest
lower bound of the mean entropy h(~,P)
on the set of all Borel finite partitions which are strongly of order 6 and whose atoms have a null b o u n d a r y for ~.
Proof.
Entropy calculations use only m o d u l o 0 partitions.
thus shows that the Borel partitions w h o s e atoms have null boundaries
upper
Proposition.
that are strongly of order 6 and
for ~, are dense for the entropic dis-
tance in the set of all partitions
5.3.5.
Remark 5.2.14
strongly of order 6.
As a function on M(X,T), m e a n entropy of order 6 is
semi-continuous.
Proof. This function is less than or equal to the greatest lower bound of the m e a n entropies of all Borel partitions
strongly of order ~ whose
atoms have a null b o u n d a r y for ~. This
last function is a greatest lower bound of continuous functions at
the point ~, and thus an upper A c c o r d i n g to lemma 5.3.4, --> h(~,6)
semi-continuous
these two functions
is then upper semi-continuous
function at ~. agree at ~. The function
at ~ because it is equal to
an upper semi-continuous function that e v e r y w h e r e bounds it above.
5.3.~.
Proposition.
The mean entropy h(~)
is the least upper bound of
all m e a n entropies of order 6.
Proof. The following inequality clearly holds for every 6 in W: h(~,~)
__<
h(~)
If P is a Borel p a r t i t i o n such that
h(~,P)
>
h(~)
- e/2
and if 0 is an open cover of X consisting in open n e i g h b o r h o o d s atoms of P, such that the overlap ratio R(~,O)
of the
is less than ~/2, we have,
for every partition Q adapted to O,
h(~,Q)
>
h(~)
W h e n ~ is an open cover, w i t h the partitions
-
e
the mean entropy of order 6 can be calculated
strongly of order 6 and adapted to the cover.
84 According
to remark 5.2.16,
for every atom Pi of P, the Cartesian square
of the closure Pi is included
in 5; and the cover by the Pi is finer than
the cover 5. It is then possible adapted 5.3.7.
to construct
to the covering. Remark.
Hence
The set M(X,T)
a subpartition
strongly of order ~ and
the result. of all invariant
Radon probability measures
on X is a simplex and every element ~ in it can be uniquely decomposed a gravity
as
center =
f
v
de
(v)
M(X,T) where e M(X,T) i.e.
is a Radon probability measure that give all the mass
on the convex and compact
to the subset of extremal
to the set of ergodic measures;
points
set
of M(X,T),
this is the ergodic decomposition
of ~. Entropy
is an affine
ergodic
decomposition holds
h(~)
function on M(X,T);
=
we can then suppose
for the entropy,
h(v)
/
de
that the
i.e. that
(~)
M(X,T) Thanks
to the previous
is an affine upper decomposition
results
on the mean entropy of order 5, h(.,6)
semi-continuous
function and the following
ergodic
holds for it:
h(~,6)
=
f
h(v,~)
de
(v)
M(X,T) We then get, according f
h(~)
de
* M(X,T)
to the very definition of the inner integral
(v)
>
Sup f
~
The inner integral
h(v,~)
of h(v)
functions, h(~)
>
(v)
=
h(~)
is therefore bounded below by h(~).
If we knew that entropy were a greatest continuous
de
M(X,T)
lower bound of affine lower semi-
we would get the inequality f
h(v)
dO
(v)
M(X,T) showing
that the entropy function
and that the ergodic decomposition We could achieve variational
is measurable holds
for all probabilities
for it.
this result as a corollary of the following
principle.
converse
85 5.3.8.
Conjecture:
entropy
is finite,
converse variational
the entropy of a measure
bound of the difference upper
semi-continuous
5.3.9.
Remark.
for instance
principle.
(p2(f)
~ is the greatest
- ~(f)) when f ranges
functions
lower
in the set of all
on X.
When the entropy function
is upper
in the case of closed invariant
the converse variational
If the topological
semi-continuous,
subsets
of symbolic
principle holds with continuous
as spaces,
functions.
5.4. REFERENCES
Adler,
Konheim and M c A n d r e w
cal ,entropy. Its relation formation Bowen
by Dinaburg
(I) have introduced
to the metric
(i), Goodman
the concept of topologi-
entropy was done for one trans-
(i) and Goodwyn
(i).
(i) should also be cited for his contribution
The pressure
of a continuous
the frame of statistical
to this study.
function was first defined by Ruelle
mechanics
and by Walters
(i) in
(I) in the general
case.
For the second part of the variational
principle,
given here was inspired by Misiurewicz
(i); we adapted his proof to the
case of pavable an smenable
groups
group
Consult Jacobs
in our work
(I) as regards
the proof of the ergodic
is Z and when the compact
The ergodic decomposition Denker,
(3) to reach the general
proof for
(6).
entropy when the group of integration
the proof that we have
Grillenberger
principle when entropy
and Sigmund is upper
functions
for instance,
(I) proved
is a result
in Meyer
the converse
semi-continuous
of
space is metrizable.
of affine semi-continuous
theory that you can find,
decomposition
on M(X,T).
(I).
variational
6. STATISTICAL MECHANICS
6.1. LOCAL SPECIFICATIONS
6.1.1.
Definitions.
anics on a lattice,
AND GIBBS MEASURES
In order to study some problems consider
configurations
compact metrizable Moreover
and, at every point of S, the set I
is finite.
tions of the system is the product Endowed with the product
topology of discrete
X is then a
to select a closed subspace of
on the configurations
stucture
is generated by the basis
=
topologies,
space.
X to take some constraints
VA
The space X of all configura-
space I S .
in certain cases it is necessary
The single uniform
in statistical mech-
the following model of the situation.
The set S of all sites is countable of all possible
ON A LATTICE
corresponding
into account.
to the compact topology of X
(VA,AeF(S)) , where V A is the subset of X 2
{ (~l,~2)eX 2, V x e A ~l(X)=~2(x)}
We then denote by VA(~)
the neighborhood
closed and open set of all configurations
of order V A of ~, i.e.
the
~' that agree with ~ on A.
On the other hand,
given a finite part A of S, A(~)
all configurations
that agree with ~ at all points
is the finite set of of the complementary
part of A. To simplify matters we only shall consider models with two elements
in I
which we will denote by +I and -I. The configuration
space X = {-l,+l} S is then a compact
The normalized Haar measure measure
group.
of this group is the product probability
h = (I/2,1/2) S
Denote by a consider,
topological
x
the coordinate
function at x given by
Ox($)
= ~(x)
and
for every finite part A of S, the product °A of all Ox where
x runs in A.
87 The real
functions
OA, where A runs in the set of all finite parts
are the characters It is possible uous
function
of the compact
f(A)
6.1.2.
group X.
here to use the Fourier on X, its Fourier =
~ f(~).OA(E) X
Proposition.
of S,
transform.
transform
If f is a real
f is defined
contin-
by
dh(~)
If f is a continuous
function
on X, the partial
sums
fA' defined by fA
=
I f(B).o B BoA
uniformly
converge
to f when the finite part A simply
Moreover,
the mapping
uous when the uniform structure Proof.
A --> fA' from F(S) structure
on P(S) which gives
Let us calculate fA(~)
Here n stands
to C(X),
is induced
to P(S)
is uniformly
contin-
on F(S) by the single uniform
its compact
topology.
=
[ {f f(~).CB(~) BcA X
dh(~)}.OB($)
=
f ( ~ OB(~).OB(E).f(~)) X B~A
=
[ ( ~ OB(n).f(nA$)) X B~A
dh(~)
dh(n)
configuration
~A$ of ~ and $ in X. The no-
tation A for the product
is of course related
of X with
endowed with the symmetrical
the group P(X)
of A, is equal
to S.
fA($)
for the product
A quick computation
tends
shows
to the natural
isomorphism
difference.
that the sum of the functions
OB, over all parts
to
21 A1 . IVA(T ) Here the symbol T stands The number
fA(¢)
mass
VA(¢);
from 0 because
for h.
of X.
is then the mean value of the function
h on the neighborhood different
for the unit element the Haar measure
every non-empty
f for the measure
of this neighborhood
is
open set has a strictly positive
88 Thanks tends
to the uniform
continuity
of f, fA uniformly
tends
On the other hand,
for every pair
(fA)A '
=
(A,A')
of finite parts
a continuous
to gA is a positive contraction of C(X). Then, for every positive real number E, there exists S such that Suppose
llf - fA, l[ ~ E
whenever
now that the two parts
intersection Denoting
A' contains
function
g on X
a finite part A of
A.
A{ and A~ agree on A, i.e.
that their
with A is the same set B.
by A'7 the union A~OA
IIfA~
fA~ll
If B' is the set AINA~,
[IfAy~B, The intersection corresponding
-
~
we get
2~
the contraction
fA~B,II
<
A local
continuous
all finite parts
property
of the proposition
specification
functions
leads
to
2g
of B' and A is B; each part A ~ B '
A~ and the proof i
Definition.
negative
of S, we get
fAQA'
and, given a finite part A of S, mapping
6.1.3.
to f when A
to S.
on X which
of S, and which verifies
is then equal
is a family ~ = (~A) of nonis indexed by the set F(S) of the following
three conditions:
a)
the function ~@ is equal
b)
for every finite part A of S and every element $ of X,
~'e (~)~A($ ') c)
for every pair
(A,A')
to the
is complete.
to I
=
1
of finite parts
of S, where
A' contains
and every element ~ of X,
~A, (¢)
If we assume normalization
=
~A(~).
that the functions condition
~ ~A, (~') 'eA(~ )
~A are not identically
(b) results
from t h e coherence
equal
to 0, the
condition
(c).
A,
89 6.1.4.
Proposition.
ponding
Given a local
family H = (H A ) of operators
IIA(f) (¢) All operators operator
=
~A
¢'e (¢
H A are positive
~@ is the identity;
commutation
and, notably,
~
)~A(¢')'f(E')
contractions
of the normed
these operators
space C(X),
fulfill
and
the
=
to verify
Now suppose
condition
consider
to s(f).
(b) of a local
H A is a positive
This
contraction,
function
inequality
follows
f on X, from the
specification.
in A'; in order
to prove
the commutation
the identity
HAO~A, (f) (~) summation
=
~ ~A(¢') ( ~ ~A, (¢") .f($") ¢'eA(¢) ¢"eA' (~')
can be factorized
HAOH A ,(f)(¢)
In the other
]IA,
for every real continuous
that A is contained
This double
=
that the operator
that,
is less than or equal
normalization
HA,OH A
they are projectors.
In order
relations,
the corres-
on C(X) given by
moreover,
HAOIIA,
we have only to verify S(~A(f))
7, consider
relations
AcA'
Proof.
specification
in the following
=
¢ 's ~(¢ )~A($')(
=
I.~A, (f) (¢)
direction,
we get
]IA,O]IA(f)(¢)
=
"e~'
)
way:
(¢)~A '(¢")
"f($")
)
~ ~A,(¢')( ~ )~A(~").f(¢ '') ) 'eA'(¢) ¢"eA(¢' ¢"eA'(¢)
f(¢") • ~ .... ~A, (¢ ') -~A(¢") ¢ 'eA(¢ )
~A, (f) (¢) The commutation 6.1.5.
relations
Definition.
between
Given a local
is a Radon probability
measure
the operators specification
H A are then established. ~, a Gibbs measure
~ on X which verifies,
part A of S and every continuous
function
f on X,
for
for every finite
90
(~A(f)) This means
=
~ (f)
that the local specification
version of the conditional
~ provides
probabilities
events
only depending on the coordinates
6.1.6.
Proposition.
The set G(~)
convex and compact
of
out of finite parts of S.
consisting
given specification ~ is non-empty.
us with a continuous
for ~ given the o-algebras
of all Gibbs measures
For the weak* topology,
for a
it is a
subset of the set of all Radon probability measures
on X; and it is a simplex. Proof.
Consider
mappings
the family
of the compact
(~A' AeF(S)
set K(C(X),s)
) of continuous
linear one-to-one
of all Radon probability measures
on X. A Radon probability measure ~ verifies u = u o~ A if and only if it belongs to the image by ~A of the convex and compact set K(C(X),s). , The images ~A(K(C(X),s)) are compact sets with the non-empty section property due to the commutation The elements precisely
of the non-empty
Theorem.
sets are
is convex and compact for the weak*
topology
Proof.
to remark
For every element
Sup ~(f) ~eG(~)
=
f of C(X),
the following
For every Gibbs measure ~, for every f in C(X),
<
greatest
inequality
and for every
is true:
is then less than or equal to the
lower bound of all S(~A(f)).
On the other hand, sublinear A tends
equality holds:
S(KA(f))
The least upper bound of all ~(f)
Then,
1.4.6.
Inf S(HA(f)) AeF(S)
finite part A of S, the following
~(f)
the operators.
for ~.
and also a simplex according 6.1.7.
between
of all these compact
the Gibbs measures
It is quite clear that G(~)
relations
intersection
finite inter-
because
the function on C(X), given by ~(f) this greatest
lower bound is also a directed
to S (due to the commutation
the sublinear
functionals
on C(X)
= Inf S(~A(f)) , is
relations
between
limit when
the HA).
function ~ is the least upper bound of all linear that it bounds
above.
91
It must still be shown that a linear functional above by the function ~ is a Gibbs measure First,
notice that the sublinear
the sublinear
function
function ~. A linear
on C(X) which
s is greater
functional
by function ~ is a Radon probability measure
Therefore,
~ which is bounded above function f of C(X),
is equal to 0 whenever
A' contains
the function ~ is equal to 0 for all previous
probability
~.
than or equal to
on X.
For every finite part A of S and every continuous the expression ~A,(f - hA(f))
is bounded
for the specification
~ is equal to 0 for these increments
A.
increments.
and so belongs
The
to G(~).
The proof of the theorem is now complete.
6.2. COCYCLES AND QUASI-INVARIANT
6.2.1.
Definition.
MEASURES
Given a finite part A of S, the one-to-one mapping
from the space {-I,+I} S onto itself
~A
is defined by
IA(X) • A(~)(x) The mapping
=
(-I)
.~(x)
rA is then a homeomorphism
in reversing
the configuration
of X = {-I,+i} S, and it consists
at all points of A; we call z A a finite
modification. The set ~ of all finite modifications The group ~ is Abelian;
it is generated by the rx' where x runs in S,
and this group is naturally
isomorphic
of S endowed with the symmetrical 6.2.2.
Definition.
space C(X)
is a locally finite group.
A cocycle
that verifies,
to the group of all finite parts
difference.
is a mapping V from the group
for every pair
(A,B)
~
to the
of finite parts of S,
the identity
VAA B A cocycle
V A + VBOr A
is then completely
in S, provided responding
=
determined by the functions Vx, where x runs
that these functions
to the algebraic
satisfy the following relations,
relations
between the generators
for every element x of S,
Vx + V x ° ~ x
=
0
for every pair
V x + VyoTx
=
Vy + Vxo~y
(x,y),
cor-
of the group:
92
6.2.3. Definition. A Radon probability measure on X is said to be quasi° invariant with respect to a given cocycle V if, for every finite part A of S and every continuous function f on X, the following equality holds: ~ (fo~A)
=
~ (f.exp (VA))
By an easy induction argument the verification of the previous equality can be restricted to the parts that consist of a single point. It is possible to state directly that there always exist quasi-invariant measures for any given cocycle. We do not demonstrate this result since the next proposition shows the link between cocycles and positive local specifications on the one hand, and between quasi-invariant measures and Gibbs measures on the other. 6.2.4. Proposition. Let V be a continuous cocycle. For every finite part A of S, define a function ~A by ~A
=
i/( ~ exp(VB)) BoA
The family ~ = (~A) is a local specification; moreover, all functions ~A are everywhere strictly positive. Proof. First, it is clear that
~@ = 1
In order to verify the normalization relation, calculate the sum , ~ )~A(~ ') eA(~
=
~ I/( ~ e x p ( V B ( ' ) ) ~'e~($) BoA
Because the configurations $' of A(~) are exactly the ~B(E) where B runs in the set of all parts of A, and according to the cocycle relations, we get
'e~A(~ )~A (~ ')
BaA
(I/( ~ exp(VcoTB(~)))) CcA
BoA
(exp(VB(~))/( ~ exp(VcAB(~)))) CcA
and the sum is equal to i. In order to verify the coherence relations, in A':
suppose that A is contained
93
¢'e
(~ )~A, (¢,)
=
~ (I/(B~A,exp(VB(~')))) ~'e (~)
=
C~A(I/(B~A,exp(VBoTC(~))))
Apply the cocycle relations to get
¢'e ($
)~A,($,)
=
With the coherence relations, hold for ~.
~A,(~)/~A(~) all properties of a local specification
6.2.5. Proposition. Let ~ be a positive local specification, continuous functions ~A are everywhere strictly positive.
i.e. all
The function Log(~A,O~A/~A,) does not depend on the finite part A' whenever this part contains A. Denote this function by V A. The family of all V A is then a cocycle. Proof.
First verify that V A is well defined by calculating
:A,OTA($ )
=
,AO:A(: ). (
~ :A,(: ')) 'eA(:A(~ ))
~A'(~)
=
~A(~)'( ~ ' e ~A(~ )~A'(~'))
Because the sets A(~) and A(~A($)) are the same, the ratios ~A,O~A/~ A, and ~AOTA/~A are equal; hence the existence of the function V A. To verify the cocycle relations, A and B and calculate VAA B VA~ B
take a finite part A' containing both
=
Log(~A,O~AO~B/~A,)
=
Log(~A,OTA)
=
VA
+
+ Log(~A,O~BO~A/~A,OT A)
V B =T A
6.2.6. Proposition. The mappings ~ --> V(~) and V - - > ~(V), defined in the two previous propositions between positive local specifications and cocycles, are inverse of each other. There is no difficulty in calculating the proof of this outcome.
94 6.2.7.
Proposition.
the Gibbs measures
Given a pair
(V,~), where V = V(~)
and
for ~ are exactly the quasi-invariant
~ = ~(V),
measures
for
the cocycle V. Proof.
Let us first prove that a Gibbs measure
for V; therefore,
we have to show that,
for ~ is quasi-invariant
for every element
g in C(X) and
every point x of S,
~ (g.exp (Vx))
=
~ (go~x)
Because p is a Gibbs measure
~(f)
=
for ~, we get
~(~x(f))
=
~((f + fOTx.exp(Vx))/(l
+ exp(Vx)))
Then
p((f.exp(Vx))/(l Choosing
+ exp(Vx)))
=
f = g.(l + exp(Vx)) , we get the quasi-invariance
In order to prove that a quasi-invariant measure
p((fo~x.exp(Vx))/(l
measure ~ for V is a Gibbs
=
~(( ~ fOTB.exp(VB))/( BaA
local
Remark.
specification
B~A ((f°~B)/(CcA [ exp(Vc°~B))
=
~ BcA
=
p (f)
((f.exp(VB))/(
of the Gibbs measures
that a Gibbs measure
a mass which is proportional
are interesting:
probabilities;
and the
of the energy
(we shall
of some measures
under the
to prove a result of Ruelle's).
Given a cocycle V = (V A) on X, there exists
J = (J(A)) of real numbers
the
gives to the configurations
to the exponential
the quasi-invariance
group of modifications
Proposition.
~ exp(Vc))) CCA
involves given conditional
means
use in the next chapter action a wider
~ exp(Vc))) CcA
=
Two properties
quasi-invariance
6.2.9.
relation.
for ~, let us simply calculate ~(~A(f))
~(HAf))
6.2.8.
+ exp(Vx)))
a family
indexed by the finite parts of the set S, such
95 that the Fourier
transform
of the continuous
function
V
is given by X
Proof.
ix(A)
=
-2J(A)
if xeA
ix(A)
=
0
if
Using
Fourier
the cocycle
coefficients
x~A
relations
Vx(A)
V x + VxO~ x = 0, we discover
are equal
that the
to 0 when the finite part A does not
contain x. The other
cocycle
that contains
relations
ix(A)
the existence
Imagine
Fourier
=
Vy(A)
ix(A)
-
= Vy(A).
of the family J, the "-2" being related
that there exists coefficients
the increments
Notice
for a part A
to the
interpretation.
the Fourier with
Vy(A)
-
that is to say to following
, lead,
both x and y, to the equality
ix(A)
Hence
V x + Vyo~ x = Vy + Vxo~y
coefficients that J(@)
E representing
-2J(A)
a cocycle
You would
and that on X
then find the
that we have just built.
function,
of energy
the energy,
and then build
V A = Eo~ A - E of the energy.
is not determined,
tation of an energy the differences
a function
of E are J(A);
which
is coherent
with our interpre-
only defined up to an additive
are actually
An interaction
constant;
significant.
6.2.10.
Definition.
indexed
by the set of all finite non-empty
is a family J = (J(A)) parts
of real numbers
of S, such that, for
every point x of S, the mapping
A --> J(A).ixe A is the Fourier
transform
These functions 6.2.11.
Definition.
if J(A)
is equal
6.2.12.
Definition.
J(A)
is equal
of a continuous
then verify
the cocycle
An interaction
function
on X.
relations.
is said to satisfy
Ising's
condition
to 0 for every finite part A with an odd cardinal An interaction
to 0 whenever
number.
J is said to be a pair interaction
the cardinal
number
of A is greater
if
than or
96 equal to 3. Notice that a function J on the set of all finite parts of S which verifies this condition,
is an interaction if and only if, for every point x
of S, the series J({x,y}) where y runs in S converges. This results, 6.2.13.
for instance,
Definition.
from proposition 6.1.2.
An interaction J is said to be attractive
if the
function J is non-negative. A non-negative function J on the set of all finite parts of S is an interaction if and only if, for every point x of S, the series
(J(A),xeA)
of real numbers converges. To prove this equivalence,
apply proposition 6.1.2 to the unit configu-
ration.
6.3. PHASE TRANSITIONS
6.3.1. Remark.
Certainly,
one of the most important problems
tical mechanics on a lattice is the phase transition problem,
in statisi.e. deci-
ding whether there are many different Gibbs measures for a given specification. When J is an interaction,
~J is also an interaction for every positive
real number 8. For some types of interactions, the comparison theorem:
it is possible to state
if there is a phase transition for BJ, there is
also a phase transition for ~'J~ whenever B' is greater than B. On the other hand,
the Kirkwood-Salsburg
little further on, postulates
theorem,
that we shall prove a
that in some cases there is no phase
transition for B adequately minute. Therefore, verifies orems,
the specific phase transition problem for an interaction which
the hypotheses of the comparison and the Kirkwood-Salsburg
the-
is knowing if there are several Gibbs measures for ~ adequately
large. Classical
interactions,
like attractive pair interactions,
verify
all these hypotheses. The problem of phase transition is a sizable one and we only give here the general 6.3.2.
theorems used to simplify this particular work.
Definition.
of all continuous convergent.
Consider the vector subspace A(X) of C(X) consisting functions on X whose Fourier series is absolutely
97
Define
then a real
function
]IIflll This
function
=
is less
bound of the norms
Proof.
measure
function
lity ~(f.(l + th(Vx/2))
V consists
of functions
and if the least upper quasi-
measure
on X that verifies
f + f°~x = 0, the equa-
=
~(f.(2exp(Vx/2)/(exp(Vx/2)+exp(-Vx/2)))
=
~(f.2exp(Vx)/(l
=
~(2fo~x/(l
-
then an operator
(I/]AI).
K(1)
=
0
Because
the °A generate and continous
the iden-
on the elements
OA
~ oA.th(Vx/2) xeA
a dense
subspace
on A(X) whenever
l[Ith(Vx/2) lll is finite.
verifies
(A,x) where x is in A.
K on A(X) by its values
=
+ exp(Vx)))
~(V) necessarily
= 0 for every pair
K(o A)
defined
+ exp(Vxo~x)))
0
for the specification
+ th(Vx/2)))
+ exp(Vx)))
~(2f.exp(Vx)/(l
=
norm of the operator
for V. For every point x of S,
this measure:
~(f.(l + th(Vx/2))
Define
of the norms.
= 0 holds.
we can calculate
tity ~(OA.(l
of A(X),
a subalthat the
for V.
Let ~ be a quasi-invariant
A Gibbs measure
to the product
If the cocycle
are elements
is, moreover,
]II.lll, meaning
lllth(Vx/2)II I is less than I, there is a single
and every continuous Indeed,
for the norm
than or equal
theorem.
V x such that all th(Vx/2)
than the supremum norm on C(X),
for it. Space A(X)
and a Sanach algebra
Kirkwood-Salsburg
invariant
it is finer
is complete
norm of the product 6.3.3.
I If(A) l AeF(S)
lll.III is a norm;
and the space A(X) gebra of C(X)
on A(X) by
If the condition
K is strictly
of A(X),
the operator
K is well
the least upper bound of the norms of the theorem
less than i.
is fulfilled,
the
g8
The restriction verifies
of ~ to the subspace A(X)
the equation
The operator
(I + K*)(~)
is a linear functional
(I + K*) has then an inverse
6.3.4.
of X.
in the dual space of A(X); and
there is only one solution for the previous thus,
that
= h, where h is the Haar measure equation
in this space and,
in the set of measures. Corollary.
are elements
If the continuous
of A(X),
functions V x that define the cocycle
and if the least upper bound of their norms in this
space is less than 7/2,
there is no phase transition
for this cocycle.
For every element ~ of A(X), whose norm is less than i, we can define th(~)
as the sum of a power
series;
ded by tg(lll~Ill). The conclusion 6.3.5.
Example.
Let us describe
S = Z n and the attractive
The interaction
and we can calculate
depend on x because When n is equal
constant)
J is equal
It is clear that the functions A(X),
the Ising model.
pair interaction
(where J is a strictly positive the lattice.
and the norm lllth(~)IIl is then boun-
is clear.
the norm
the interaction
6.4.1.
cocycle belong to invariant. immediately
in the Ising model
confirms
in dimension i for
B.
llIth(Vx/2)lll = 2 th(2J),
that there is no phase transition when th(2J)
which
shows
is less than I, i.e. if J
(Log3)/4.
6.4. SUPERMODULAR
INTERACTIONS
Definitions.
Denote by ~ the order on the product
nl ~ n2
t. .~
in
lIIth(Vx/2)III ; this norm does not is translation
any v a l u e of the inverse temperature is 2, we get
= J
to 0 otherwise.
to i, lllth(Vx/2)lll = th(2J), which
When the dimension
the lattice
for the pairs of neighbors
of the corresponding
that there is no phase transition
is less than
Consider
J defined by J({x,y})
VxeS,nl(X)
set X = {-I,+I} S defined by
=< n2(x )
99
It is the product
order of all natural
orders
of the factors;
it is not
a total order. A continuous decreasing
function
for this order.
monotonic
continuous
functions
belong
6.4.2.
V x + VyOT x = Vy + VxOTy
function ~
the cone of all these
on X; to be specific,
Given two different
VxoTy - V x This
Let us call then M(X)
functions
if it is non-
all coordinate
to M(X).
Definition.
relation
f on X is said to be non-decreasing
xy
=
points
x and y of S, the cocycle
leads to the equality
VyOT x - Vy
has the following
=
~xy
Fourier
transform:
A
~xy(A)
=
4J(A) . IxeA. lyeA
When all functions
+xy.Ox.Oy
to be supermodular.
This
are non-negative,
is the same as saying
of S, the function V x is non-decreasing 6.4.3.
Example.
A typical
vided by an attractive
example
the interaction that,
for every point x
on the subset {o x = -i} of X.
of a supermodular
pair interaction.
is said
interaction
is pro-
The function Sxy then reduces
to 4J({x,y})OxOy. 6.4.4. modular
Proposition.
Let A be a finite part of S and # be a local
specification.
that n 1 ~ n 2. Denote by ~+n
Let n 1 and n 2 be two elements
the configuration,
which
tion of the element
$ of {-i,+i} A and of the element measures
on {-i,+i} A, verify
the Holley relation:
for every pair
#l(~iV$2).~2(~lh$2 )
Proof. product
~ --> ~A($+nl)
(EI,$2)
We have to prove,
~
of {-I,+I} S A such
is the result
The two probability
of elements
super-
of the concatena-
n of {-I,+I} S A
and ~ --> #A(~+n2),
defined
of {-i,+I} A
#i(~i)-~2(~2 )
for every pair
(~i,~2)
of elements
of the
space {-i,+I} A, the inequality
~A((~IV~2)+nl ).~A((¢IA~2)+n2 )
>
#A($2+n2 ).~A($1+nl )
1O0
This
inequality
is equivalent
to
~A( (~ I W 2)+n i)/~A(~ l+n 1 ) According equal
to proposition
6.2.6,
to exp(VB(~l+nl)) , where
is greater points
than $1(x).
at which ~2(x)
The right-hand
= +i and $1(x)
to be demonstrated
the value
This results
B is the subset is equal
is supermodular;
Theorem.
product
[~,~
to the product
or-
of X of all configurations
of B.
of B in an arbitrary
Then, 7' is greater is non-decreasing
~,~
and V B is also non-decreasing
Let 7' and 7" be two probability
and that verify
order.
on the subset
space {-I,+I} A which give a positive
this set,
to exp(VB((~iA$2)+n2 ) .
that, with respect
term in the sum is non-decreasing
6.4.5.
of A of all
Vbl + Vb2°~ b I + • • + VbnO~bl o • • oT bn_ 1
=
where bl, .... b n are the points interaction
is
= -I.
on the subset
-i at all points
side of the inequality
for the part of A on which ~2(x)
from the decomposition
VB
Every
the left-hand
In other words,
der, V B is non-decreasing
~ A(~ 2+n 2)/~A( (~ IAE 2)+n 2 )
B stands
side of the inequality
It then remains taking
>
the Holley
than or equal
for the product
mass
relation
to 7",
measures
because
the
on [ ~ , ~ . on the finite
to every element
of
(6.4.4).
i.e.
for every function
order, ~'(f)
is greater
f which
than or equal
to ~"(f). The proof of this result
can be found in Holley
6.4.6.
to the previous
Corollary.
Thanks
the least upper bound of ~A(f) is equal 6.4.7.
to 1 at every point
of ~
= HA(f)(T) , weakly
is a Gibbs measure +
(f)
=
results,
when f belongs
at the configuration
to M(X),
T, which
of S.
Proposition • The family
where ~ ( f )
is reached
(I).
(~+A ) of Radon probability converges
when A tends
for ~ and verifies,
Sup ~ (f) eG(~ )
measures
on X,
to S. The limit ~+
for every f in M(X),
101
Proof. When f belongs functions
to the cone M(X) of all continuous
on X, the following
~A(f) (I)
=
equality holds
non-decreasing
for every finite part A of S:
S(~A(f))
And this directed family of real numbers has a limit when A tends to S; this
limit is the least upper bound of all ~(f), where ~ is a Gibbs
measure
for ~.
This family of probability measures ments
(fl
then simply converges
for all ele-
f2 ) of C(X), where fl and f2 are in M(X).
The subspace,
consisting
of these differences,
is dense in C(X)
uniform norm and the family is equicontinuous.
for the
Hence the existence
of
the weak limit ~+ is proven. For every f in M(X), ~+(f) this
infimum is equal
= Inf(s(~A(f));
according
to theorem 6.1.7,
to the least upper bound of all ~(f), where ~ is
in G(~). + . In order to verify that p is actually a Gibbs measure for =, notice + + simply that ~A(~B(f)) = ~A(f) whenever A contains B, and take the limit when A tends to S. 6.4.8. weak
Remark.
The probability measure ~- is similarly defined as the
limit of the ~A(.)(---T). It gives the least upper bound of all Gibbs
measures 6.4.9.
for the non-increasing
Proposition.
specification
continuous
functions.
There is a phase transition
if and only if the two particular
for a supermodular + measures ~ and ~
local are
different. Proof.
The condition
is obviously
sufficient because ~
and ~
are Gibbs
measures. If they agree,
proposition
6.4.7 and remark 6.4.8 show that a Gibbs
measure
for ~ takes necessary
because
this subspace
Gibbs measure 6.4.10.
values on the subspace
is dense in the uniform norm,
for ~.
Proposition.
is a phase transition
Let ~ be a local
The condition
is also necessary,
supermodular
specification.
There
for ~ if and only if there exists a point x in S
such that u+(~x ) is strictly greater Proof.
generated by M(X); there is only one
is obviously
than ~-(~x ).
sufficient.
we have only to demonstrate
In order to prove that it the following
inequality:
102
(u+ - ~-)(pA ) where PA is the positive
PA In this case,
=
<
(1/2). ~ (~+ - u-)(~x ) xeA
function
defined
~ ((°x+l)/2) xeA
the two Gibbs measures
~+ and ~- agree on all OA; because
the set consisting
of all PA generates
measures
are equal
and there is no phase
previous
proposition.
Let us then demonstrate
By an easy induction inequality,
<
subspace
transition
of C(X),
according
relation
~ (u+ - u-)(px)
argument
we can deduce
this inequality
which holds when the two finite parts
is equivalent
the two
to the
xeA
(~+ - ~-) (PAUB) This relation
a dense
the subadditive
(~+ - ~-)(pA )
following disjoint:
by
=<
from the A and B are
(~+ - ~-) (pA) + (~+ - ~-) (pB)
to
+ (PAUB - ~A - PB + i) This which uous
last inequality is equal function
6.4.11.
describing
holds because
to the product
interactions
however,
belongs
to M(X)
contin).
we shall turn our full attention
which we have just anticipated
we can say that the difference
on the point x when the interaction
the action of a geometrical
to
while
of one another
(~+ - ~-)(o x)
is invariant
under
group.
when the interaction
has the Ising property,
by the involution
Then,
the nullity
of the non-negative
rizes
the absence
of phase
articles
(PAUB - PA -PB + I),
the Ising model.
does not depend
the images
- OA - PB + i)
- pB ), is a decreasing
-(l-PA)(l-PB)
In the next chapter,
invariant
At this point,
Moreover,
the function
(I - pA).(l
on X ( the function
Remark.
translation
=< ~-(~AUB
number
transition.
that deal with this number
• which reverses
all spins.
~+(o x) completely
The numerous
call
~+ and ~- are
mathematical
it spontaneous
charactephysics
magnetization.
103
6.4.12.
Proposition.
local supermodular compact
The Gibbs measures
~
and ~ , relative
specification 7, are extremal
to a given
points of the convex
set G(7) of all Gibbs measures.
Proof. Measure ~
+
agrees on the cone M(X) with the function 7, which is
the least upper bound of all Gibbs measures sible to decompose
for ~. It is therefore
this measure because this cone generates
impos-
a dense sub-
space of C(X). A similar result holds for 6.4.13.
Remark.
not sufficient
The previous to postulate
hold for attractive 6.4.14. nite;
results
about supermodular
a comparison
theorem.
are
theorems
interactions.
Proposition.
Let X A be the product
set {-I,+I} A where A is fi-
and let E 1 and E 2 be two real functions
forms verify,
interactions
The comparison
on X A whose Fourier
for every part B of A (except perhaps
trans-
the empty part),
the
relations
Then,
for every character,
are proportional
the probability
to the exponentials
~l(OB)
measures ~I and ~2 on X A which
of these functions
__< ~2(OB )
The proof of this result can be found in Preston 6.4.15. that,
Proposition.
~
(I).
Let Jl and J2 be two attractive
for every finite part A of S, the following
Jl (A)
proposition Proof.
6.4.16.
such
J2 (A)
and ~2,A(.)(T),
probabilities
verify the order relation described
on XA,
in the
6.4.14.
A quick computation
the outside configuration proposition
interactions
inequality holds:
Then, for every finite part A of S, the conditional 71,A(.)(T)
are related by:
shows that these conditional understood
energies,
as T, satisfy the hypothesis
with of
6.4.14.
Corollary.
Let J be an attractive
Ising pair interaction.
Then,
104
if there is a phase transition
for some BJ, where B is a positive
real
number,
there is also a phase transition
greater
than B.
Indeed,
the interactions ~J and B'J verify the hypotheses of proposition + and the inequality holds at the finite rank between the ~A(Ox)
6.4.15;
for the interactions
for every B'J, where 6' is
BJ and B'J. The limit inequality
therefore
implies
the corollary. 6.4.17.
Remark.
If J is an Ising attractive
all values of the positive real parameter transition
is a non-empty
may be finite or infinite.
interval
pair interaction,
the set of
B for which there is no phase
[0,Bcl . The non-null
limit value
Bc it may or may not belong to
If it is finite,
the set. 6.4.18.
Remark.
For a given Ising attractive
pair interaction,
the phase
transition problem is deciding whether B c is finite or not, and, eventually,
calculating
this value when it is finite.
For the Ising model
in dimension 2 for instance,
devised by Peierls,
it is possible
the spontaneous magnetization
thanks to an argument
to show that for B adequately
large
is not 0.
6.5. REFERENCES
Ruelle
(I) provides
the general
ton (I), in particular, supermodular
for this chapter while Pres-
for the study of attractive
is a notion of our own and is used in our work (5).
Holley first stated his theorem
(I) by using Markov processes.
proofs were given in 1974 during the conferences
dic Theory in Rennes by Brunel
(i) and by Hansel
Phase transition
models
physics
and
potentials.
Quasi-invariance combinatorial
foundation
gives the basis
in particular
to be given specific
Direct on Ergo-
(I).
is too broad an area of mathematical
citations here.
7. DYNAMICAL SYSTEMS IN STATISTICAL MECHANICS
7.1. INVARIANT LOCAL SPECIFICATIONS
7.1.1.
Definition.
Let X be the compact m e t r i z a b l e space I S . When the
countable set S is a group G, we can consider the action of G on the space X by translations models
(described in 2.1.2)
and then study invariant
in statistical mechanics.
A local specification ~ is said to be invariant if this family verifies, for every finite part A of G and every element g of G, the coherence relation ~AoT g = 7.1.2.
~Ag"
Definition.
Let ~ be a local s p e c i f i c a t i o n on X. A Gibbs measure
for ~ is said to be invariant
if it is invariant under the action of G
by translations on X.
7.1.3.
Remark.
invariant, However,
If a local s p e c i f i c a t i o n has a Gibbs measure which is
then this s p e c i f i c a t i o n is invariant.
it is not generally true that all Gibbs measures
for an invar-
iant s p e c i f i c a t i o n are invariant.
7.1.4.
Proposition.
If the group G is amenable,
Gibbs m e a s u r e s for every invariant
Proof.
there exist invariant
local specification.
Group G clearly acts on the convex and compact set G(~) by affine
continuous
one-to-one mappings.
Since G is amenable,
there exists at
least one invariant point in this set and the invariant points are precisely the invariant Gibbs measures for
7.1.5. Definition. tion ~(V) verifies 7.1.6.
A cocycle is said to be invariant if the specifica-
is invariant.
This is equivalent to saying that this cocycle
all coherence relations VAg = VAoTg.
Remark.
The group T of translations normalizes
the group ~ of all
106
finite modifications;
and,
an invariant
to ¢ of a mapping V from the group ~ verifying
=
V a + Vboa -I
and whose restriction Definition.
ponding
(generated by T and ¢) into C(X)
the following relations:
Va. b
7.1.7.
cocycle V is the restriction
cocycle
to T is 0.
An interaction
is invariant;
is said to be invariant
this property is equivalent
if the corresto the invar-
iance of J under the right translations VgeG,VAeF(G),
7.2.
J(Ag)
INVARIANT GIBBS MEASURES
7.2.1 Remark.
= J(A)
AND EQUILIBRIUM MEASURES
The subset I(~) of all invariant Gibbs measures
for ~ is
defined in G(~) by the invariance property. We want now to define it as a subset of the convex and compact
set M(X,T)
(which can also be denoted by K(C(X),s,T)
2) consis-
according
ting of all invariant Radon probability measures The variational tion, gives 7.2.2.
principle,
when applied
such a characterization;
Theorem.
e = (ex,xeG)
Given an invariant
of continuous
to chapter
on X.
to a particular
continuous
func-
now we have to build such a function. interaction J, there exists a family
functions
on X with the following
two prop-
erties: first,
family e is coherent under the action of G on X by transla-
tions,
i.e. for every pair
and,
second,
ex(A)
Proof.
the Fourier
=
(x,g) of elements
transform of the function e
X
is
IxeA.(J(A)/IA])
Because J is invariant under
it exists,
of G, exg = exoTg;
is certainly coherent.
the translations,
such a family,
if
107
Now we have only to show that the real function on F(G) given by
A --> Ixe A.(J(A)/IAI) is actually the Fourier The Fourier The next
lemma
on X whose
transform of a continuous
function.
transform of Ox.Vx is -2J(A {x}).Ix~ A. (7.2.3)
Fourier
shows that there exists a probability measure
transform
is
A
v(A)
=
v(o A)
=
I/(I+[A I)
The function e x = -(I/2).ax.(V~(ax. Vx)) rier transform is the one given above. 7.2.3.
Lemma.
Since the Fourier
are looking for depends corresponding geable,
Fourier
transform is given by v(o A) = I/(I+IAI).
transform of the probability measure only on the cardinal
to the character,
to the permutations
this probability measure must be exchan-
is in fact necessary
according [0,~
I / 0 v k ( O A ) de(k)
as an average of power measures
(~,I-~) G. We have to find a probability such that,
=
(this
to the De Finetti theorem).
the probability measure
measure e on the segment
of X cor-
of G.
We then look for this probability Call v~
that we
number of the finite part
i.e. invariant under the action of all homeomorphisms
responding
i.e.
and its Fou-
There exists a Radon probability measure v on the compact
group X = {-i,+I} G whose Proof.
is then continuous
for every finite part of G,
I/(I+[A I)
for every natural number n, i /
(2).-1)n de(~)
=
I/(l+n)
0 The probability measure solves
the equation.
7.2.4.
Definition.
called
local energies.
2.1(~>i/2).m,
The functions
e
where m is the Lebesgue measure,
that we built
in theorem 7.2.2 are
X
transforms, energies
Formally
speaking,
i.e. as regards
Fourier
it can be assumed that the energy is the sum of the local
e x over all sites.
108
7.2.5. Theorem. Let ~ be an invariant local s p e c i f i c a t i o n on the space X = {-I,+I} G and let the group G be amenable. The e q u i l i b r i u m m e a s u r e s for a continuous function f on X are the measures for which p(f) = h(~) + ~(f). For the continuous
functions
ex, the e q u i l i b r i u m measures
are the same for
all sites of G, and are p r e c i s e l y the invariant Gibbs measures
Proof.
Because the e X constitute a coherent family,
measures
for ~.
the e q u i l i b r i u m
are the same for all of them.
Such e q u i l i b r i u m measures do exist,
according to the variational prin-
ciple, because the entropy is, in this case,
an upper semi-continuous
f u n c t i o n of the measure. We have proven elsewhere
(5) that the e q u i l i b r i u m m e a s u r e s
are the
invariant Gibbs measures.
7.3. M I X I N G PROPERTIES
7.3.1 Definition.
The asymptotic o - a l g e b r a of the compact space X = I S is
the greatest lower bound of all Borel o-algebras consisting of the events only depending on the coordinates out of a given finite part of S. This o-algebra also consists of all events that are invariant under
the
action of the group ~ of all finite m o d i f i c a t i o n s of X.
7.3.2. Theorem.
Let ~ be a local specification on X = I S. The extremal
points of the convex and compact set G(~)
are the Gibbs measures whose
r e s t r i c t i o n to the asymptotic O-algebra is trivial.
Proof. The convex and compact set G(~) has the p r o p e r t y described in 1.4.3 and its extremal points are the elements of it for which the subspace Here,
NCR. I is dense for the L l - n o r m in C(X).
N is the subspace generated by all differences
(f - [A(f)), where
f runs in C(X) and A runs in F(G). For every element ~ of G(~), of LI(X,~,~)
the operators ~A are positive
because p o s i t i v i t y extends from C(X)
For every continuous
to the whole set L I.
function f, we then get the inequality
;~(IzA(f) 1 )
=<
Z(ZA(Ifl))
=
~(Ifl)
contractions
109
and the extension of the operator On the other hand,
tends to the constant also holds result
the elements
7.3.3.
Remark.
and compact 7.3.4.
As regards
This is equivalent
to the triviality
the quasi-invariance
aspect,
of the extremal
of
the above result is points
set of invariant Radon probability measures
the abstract
this
for the given measure.
Let ~ be an invariant
an invariant Gibbs measure Then,
C(X),
of L I.
to the characterization
Proposition.
uniformly
of L I that are invariant under all operators
functions.
the asymptotic ~-algebra
quite similar
function ~A(f)
Because of the density of the subspace
is true for all functions
~A are the constant
in [email protected],
function p(f) when A tends to S; and the convergence
in Ll-norm.
In particular,
to L I is a contraction.
for every element
local
specification
for ~ which is also extremal
dynamical
system
(X,~,~,T)
of the convex
as ergodic ones. and ~ be
in G(~).
has the following
strong
mixing property: For every pair
(a,b)
of events
and every positive real number e,
there exists a finite part F of G such that the inequality l~(a~Tg(b)) holds whenever
Proof.
- p(a).~(b)I
<
E
g does not belong to F.
The partition at the unit element
of G is a generator;
we can find, for every positive real number ~, two measurable X, a' and b', depending the coordinates
therefore parts of
only on a finite number of coordinates,
in A for a' and on the coordinates
in B for b',
e.g. on such
that p(aAa')
<
e/5
and
~(bAb')
<
e/5
The following upper bound is easily calculated: Ip(a~Tg(b))
- ~(a).~(b)l
<
l~(a'~Tg(b'))
We then need only state the mixing property
for two events only depending
on a finite number of coordinates. If ~(b')
- ~(a').~(b') I + 4~/5
is equal to 0, there is nothing to prove.
110
Otherwise, the problem consists in proving that the limit, when g runs out of every finite part, of the ratio ~(a'nTg(b'))/~(b') is equal to the measure ~(a'). This results from the martingale theorem: The conditional expectation with respect to the o-algebra of all events that only depend on the coordinates out of a finite part, converges almost everywhere and in Ll-norm to the conditional expectation with respect to the limit o-algebra. The limit o-algebra is the asymptotic one; because this o-algebra is trivial, the limit of the above ratio is actually ~(a'). 7.3.5. Definition. Let (X,~,~,T) be an abstract dynamical system, and let the acting group G be amenable. The weak mixing property is said to hold for this system if, for every pair (a,b) of events, the average value (I/IAI).( ~ I~(anTg(b)) - ~(a).~(b)I) geA tends to 0 along the ameaning filter. 7.3.6. Remark. Definition 7.3.5 is clearly equivalent to the analogous statement on all pairs of elements in L2(X,~,~). The triangle inequality then shows that, for every pair (fl,f2) of elements in L2(X,~,u) , ~(fl.(I/IAl).( [ f2oT g )) - ~(fl).~(f2) geA tends to 0 along the ameaning filter. An invariant element f2 of L 2 thus verifies ~(fl.f2) = ~(fl).~(f 2) for every element fl of L2(X,Q,~); and f2 agrees almost everywhere with the constant function ~(f2 ). The only invariant elements in L 2 are then the constant functions. Thus, the weak mixing implies the ergodicity of the system. 7.3.7. Proposition.
Strong mixing, defined in 7.3.4, implies weak mixing.
Proof: elementary calculations. 7.3.8. Proposition. Let (X,R,~,T) be a dynamical system, let the acting
111 group G be amenable and suppose that the weak mixing property holds for the dynamical system. Then, for every ergodic dynamical system (Y,~,~,U), the product dynamical system (where the probability space is the product space of the two probability spaces and where the action is the componentwise action) is an ergodic system. Proof. Because of the density of the vector subspace generated by the functions f(x)g(y), we have only to prove, for every pair (fl,f2) of elements in L2(X,Q,~) that the quantity
and every pair (f3,f4) of elements in L2(Y,~,~),
(~v)((fl®f3).(I/IAl).(
~ (f2oTg)®(f4oug))) geA
tends, along the ameaning filter to ~(fl).~(f3).~(f2).v(f4). We now have to show that the following average of differences (I/IA[).( ~ (~(fl.f2oTg).v(f3.f4oug) geA
- ~(fl)~(f3)~(f2)~(f4)))
tends to 0 along the ameaning filter. Every difference in the sum can be written (~(fl.f2oTg)
- ~(fl)~(f2)).~(f3.f4oug))
+ ~(fl).~(f2).(~(f3.f4oug)
- ~ (f3) .~ (f4))
Hence the following upper bound for the absolute value of averages of differences: If312-[f412.(I/IAl) • [ l~(fl.f2=Tg) geA
- ~(fl).~(f2) I
+ l~(fl )I "l~(f2)I "I v(fB'(I/IAl )'g~A f4°Ug)e
- ~ (f3) "v(f4)[
The first term tends to 0 along M due to the weak mixing property of the dynamical system (X,~,~,T). The ergodicity of the second factor implies that the second term also tends to 0 along the ameaning filter. 7.3.9. Remark. When ~ is an invariant supermodular specification, two Gibbs measures ~+ and ~- have the strong mixing property.
the
112
Indeed,
they are extremal
Notably,
for attractive
ones in G(~) and they are also invariant.
invariant pair interactions
perty there is no phase transition
The mixing property enables us to calculate model
in dimension 2 in particular)
7.3.10.
Remark.
interaction
properties
of the dynamical
systems cor-
supermodular
than the strong mixing property.
We are interested is completely
of non-negative
numbers J(Ix-yl)
7.4.2.
If the sequence
Theorem.
in invariant
characterized
attractive
Ising pair
by the converging
series
= J({x,y}). (J(n),neN,n~2)
is sufficiently
in the sense that the series n.J(n)
phase transition condition
as
on {-I,+i} Z.
Such an interaction
decreasing,
magnetization
A THEOREM OF RUELLE'S
Definition.
interactions
(the Ising
of the correlation ~+(~x.~xg).
~÷ and ~- for an invariant
are in fact richer
7.4. EXAMPLE:
7.4.1.
The stochastic
to the measures
in some models
the spontaneous
the limit, when g tends to infinity,
responding
with the Ising pro-
if and only if ~+(o x) is equal to 0.
converges,
for J (neither for 8J, with a positive
quickly
there is no
B because the
is homogeneous).
The original
proof was done by Ruelle.
The present
the proof of this result as an illustration
section is devoted to
of the quasi-invariance
and
mixing properties. 7.4.3. Remark. equal
Consider
to E(A) = J(A).
X which consists
the "energy function" with a Fourier transform
Consider
in reversing
the homeomorphism the configurations
that are strictly less than x (for the natural The difference rx(~),
r x of the compact
order).
of energy between a configuration
$ and the configuration
is then given by its Fourier transform (Eor x - E)(A)
=
-2J(A)
=
0
if
l{yeA,y<x}l
otherwise
space
at all points y of Z
is odd
113
In the case of a pair interaction,
we get,
for y < x ~ z,
A
(Eor x - E)({y,z}) But Ruelle's increment
condition
=
indicates
then actually
-2 J(ly-zl) that this series
exists as a continuous
is convergent;
this
function on X and we are
lead to the examination
of the quasi-invariance
under
the action of a
group of homeomorphisms
of X which is wider than the group ~ of all
finite modifications. 7.4.4. r
X
Definition.
Let us give now a formal definition
of the reversal
on the left of x: rx(~)(y)
Similarly,
=
the reversal rx(~)(z)'
=
All these homeomorphisms
((_I)Iy <x ).~(y)
r X' on the right of x is defined by ((-i) Ix
of X are involutions
and commute with each
other. !
!
Moreover, the reversal rx at x is equal to rxorx+ I and also to rxorx+l; and the group U of all homeomorphisms of X which is generated by all reversals
r
and all r' then contains ~, because X
elements 7.4.5.
Proposition.
all the
the corresponding
<
the summability
=
condition
+ ®
cocycle V is the restriction
U to C(X) which verifies
Vab
system of this group.
If J verifies
n.J(n)
This cocycle
it contains
X
of a generating
the following
Va + Vb°a
to ~ of a mapping V from
cocycle relations:
-I
is completely determined
by its values
on a set of genera-
tors of U. The function V X that corresponds
Vx
=
to the reversal
~ -2J( ly-z I ).Oy.O y<x
r X is defined by
114
and the function V' x
=
In particular, the product
7.4.6.
[ -2J(ly-zl ) .o y<x~z y'°z
and for example,
=
V
+ V'or X
=
X
of all these results
Proposition.
is quasi-invariant Proof.
the reversal
T at all sites
is equal
to
rxor i and we obtain V
The proof
to r xI by
corresponding
is elementary. + u , with
The measure under
0
the action
We only have to prove,
the cocycle V as its modulus,
of U.
for every continuous
function
f on X, and
every point x of Z, the equalities + (for x)
=
u+(f.exp(Vx))
+(forl)
=
~+(f exp(~')) "
Indeed,
these reversals
We only prove it. Consider
generate
the first equality
U and the proof because
__< y
<
we get,
~+(f°~A
)
with respect
~+(f'exp(VA
n
function
of Z such that
function
f,
)) n
the continuous foT A
similar
to the action of ~ (with V
for every continuous
=
is quite
x
u+ is quasi-invariant
as its modulus)
Because
shall be achieved.
the second
then the finite part A n of all points x-n
Because
X
function
uniformly
tends
f is uniformly
continuous
on X, the
to for x when n tends to infinity.
n On the other hand,
the finite modification
product of the reversals Hence, using the cocycle
=
VA n
~A
can be written n
r x and rx_ n. relations of V, we get
Vx_ n + Vxorx_ n
as the
to
115
The quasi-invariance
property
(f.exp(V A ))
leads to
=
~+(f.exp(Vxorx_n).exp(Vx_n))
n
The uniform uniformly
continuity
of the continuous
function V x shows
We have thus proved
the following
convergence
result:
+ lim ~ (f.exp(Vx).exp(Vx_n)) n--> +® Because
that Vxorx_ n
tends to V--x when n tends to infinity.
the interaction
is invariant,
and we use now the strong mixing
=
p+(for x)
Vx_ n is the translate
property
VxoT n of Vx;
of ~+ to get
+ (for x) In order
to conclude
is equal
to i.
=
~+(f.exp(Vx)).~+(exp(Vx ))
the proof,
we still have to show that ~+(exp(Vx))
Hence we must note that the previous function, proof. 7.4.7.
and in particular
Corollary
i.e.
equal
sition
to ~-
for the constant
(proof of theorem
for the U-cocycle
equality
7.4.2).
V, it is in particular , and there
is no phase
holds
for every continuous
function
Because ~
invariant
+
i, achieving
is quasi-invariant
under
transition
the
the reversal
according
3,
to propo-
6.4.9.
7.5. REFERENCES
The characterization measures
of the invariant
for a suitable
by this author The original
continuous
proof
of the uniqueness by Ruelle
The study of attractive condition
described
invariant
way
in the last sec-
pair interactions
greater
in dimension
for which J is proportional
is then the summability
is necessarily
in a general
(5).
(i).
been done for the interactions exponent
result
as the equilibrium
is proven
in a joint work with D. Pinchon
tion was achieved
interaction
Gibbs measures
function
to n
of the series
than I. If it is greater
i has ; the
and the
than 2, Ruelle's
116
theorem shows that there is no phase transition verse temperature
stated that there is a phase transition therefore
for
a comparison with the hierarchical
= 2, known as Anderson's model, and Spencer
for any value of the in-
B. For an ~ strictly between 1 and 2, Dyson B adequately model.
large.
He used
The borderline
case,
was only solved recently by FrShlich
(I); they showed that there is a phase transition
values of the parameter.
(1,2)
for large
8. EQUIVALENCE OF COUNTABLE AMENABLE GROUPS
8.1. TILING AMENABLE GROUPS
8.1.1. Definitions. The sequence of segments in Z, (An = {0 .... n}), encountered at the beginning of this text is the prototype for all F~Iner sequences in countable amenable groups. But the elements of this sequence,
the segments An, have an interesting
property which is a decisive tool for solving some problems of ergodic theory:
they are tiling sets.
A finite part of a group,
for which it is possible to tile the whole group
with some pair-wise disjoint right translates of it, is called a tiling set. This notion is therefore in fact related to equivalence classes of finite parts under the right translations which can be called forms. Some authors,
Ornstein and Weiss in particular,
have used F~iner sequences
consisting of tiling sets in groups other than Z. Notice that, in all known amenable groups,
it is always possible to con-
struct explicit F~Iner sequences consisting of tiling sets. This means that the class of groups with arbitrarily invariant tiling sets is stable, like the class of amenable groups, under the constructions we described in section 2.3
(see Moulin Ollagnier and Pinchon
that
(3)).
But, as we said in chapter 2, no structure theorem for amenable groups is stated herein. Connes,
Ornstein and Weiss have introduced the notion of almost tiling:
with a finite number of different forms, each of them being as invariant as desired,
it is possible to cover the greatest part of the group with
almost disjoint right translates of these forms. This property is true for all amenable groups and is the subject of the next lemma.
118
8.1.2. Tiling
lemma. Let A, A and D be three finite parts of a group G
and let the unit element e of G belong Then,
there exists a partition
right
translates
of subsets of A, such that the proportion
in A for which all left translates atom,
to D.
of A, whose atoms are contained
is greater
dg by elements
Let A' be the subset of A consisting
is contained
of D are in the same
than
(I - m D ( A ) / I A I ) . ( I A I / I D - I A I ) . ( I
Proof.
in some
of all points
- m D ( A ) / I A I)
of all points
g such that Dg
in A.
By definition,
we get
IA - A'I = mD(A)
Denote by T the set of all total orders on A-~A = {geG,Ag~A~@}. Consider P(T)
the elements
gl .... gk''"
as the partition of A whose
following
A1
=
AnAg 1
A2
=
(A~Ag2)\A 1 k-i (A~Agk)\( U A i) 1
The notation P(T,x) measure
then stands
for the class of x in the partition P(~).
the uniform probability measure
~ on T, i.e. the probability
giving to every point ~ of T the mass
For a given element x in A', the probability P(~,x)
is exactly the probability
the first element This
of the
finite sequence:
"'" Ak =
Consider
of A-~A in the order T, and define atoms are the non-empty elements
I/(IA-~AI)!. that Dx is contained
that Dx is contained
in the order T such that Ag meets Dx.
last event depends only on the restriction
subset A-~D.x; probability Therefore,
in
in Ag, where g is
and its probability
of the order • to the
can be easily obtained because
for a given element g being the first in A - ~ the probability
is equal
of the event that we are searching
the ratio l{g,AgcDx}I/I{g,Ag~Dx#@}l
the to
is equal
to
119
i.e.
to the ratio
(i
Summing
mD(A)/IAI).(IAI/ID-I.A
-
I)
on all x in A', we get the expectation
E( [ iDxcP(~,x )) xeA
~
for
E( ~ IDxcP(~ x)) xeA' ' (IA'I).(IAI/ID-~AI).(I
The proportion positive
(I
There
of all x of A such that Dx is contained
function
of T whose expectation
is bounded
mD(A)/;AI).(IA I/ID-I.AI).(I
-
then exists
the above number.
an order T for which The corresponding
expected inequality. triple (A,A,D).
in P(T,x)
is a
below by
- mD(A)/IA ])
this proportion
partition
Such a partition
- mD(A)/IAI)
P(T)
in greater
thus verifies
is said to be adapted
than the
to the
8.1.3. Iterated tiling lemma. Let A, A 1 . . . . , An, and D 1 .... , D n be finite parts of a group G such that
eeD I c D 2 ~ There
then exists I) 2)
a family
for every right
... c D n (PI ..... Pn ) of partitions
subscript
translates
for every i from 1 to n-l, tion of Pi' i.e.
3)
the proportion is contained ponding
(i
Proof.
i between
of subsets
Consider
1 and n, the atoms
of Ao l the partition
Pi+l is a subparti-
the atoms of Pi+l are unions
of the points
of Pi are
of atoms of Pi
x of A for which every set D.x l containing x of the corres-
in the atom Pi(x)
partition
-
of A such that
m D (A)/i n
Pi is greater
than
i=n AI. ~ (IAil/ID?llAil).(l i=l
the finite probability
space
- mDi(Ai)/IAil)
(T,~), which
of the (Ti,~i) where T i is the set of all total orders the uniform probability on it.
is the product
on A?~AI and ~i is
120
!
For every n-uple
• = (~i ..... ~n ), consider
partitions
where every P! is built as it was in the proof of the
of h
preceding lemma. by setting
A new sequence
p
Pn-I P1
Conditions
of partitions
=
p, n
=
p,
=
Pi V P2
n
Call A' the subset of A consisting
E
i=n ~ ({TeT, i=l
where
Pi(T,x)
built
from the n-uple
stands
By construction
is then defined
of all x such that DnX is contained of the event
Dix(Pi(~,x)}) x of the partition
Pi(~),
~.
of the sequence
of all Pn from the one of the P'n' the
defined
i=n j=n ( ~ ({reT,
as the following
J
of D i :increases, definition
intersection:
D~x~P'(T,x)})) J
i=l j=i
Since the sequence
of
by such a sequence.
for the atom that contains
event E can be equivalently
get the following
(PI,...Pn)
for an x in A', the probability
=
(PI ..... Pn)
n-I V Pn
1 and 2 are then fulfilled
in A; and calculate,
the sequence
this expression
can be simplified
to
of E:
i=n E
({TeT,Dix c P~(r ,x) })
=
i=l
Because
the factors
get the probability
of the product
space T are mutually
independent,
we
of this event as the product
i=n 1 H (I - mD(Ai)/IAil).(IAil/ID.7~.Ail).,,,,, ~ , i=l l Conclude
as in the previous
A family of partitions tiling
lemma by summing
that verifies
and is said to be adapted
over all x in A'
the inequality
is called
to the given 2n+l-uple.
an iterated
121
8.1.4. Definition.
Denote by T O the subset of all total orders on G which
are isomorphic to the natural order of Z, i.e. the orders without a first element, without a last one, and for which there are only finitely m a n y elements b e t w e e n two given ones. Set T O is a subspace of the Hausdorff compact space T of all total orders on G, and it is invariant under the action of G d e s c r i b e d in 2.1.3. When the amenable group G is countable,
this set is not empty.
From now on, we shall work under the a s s u m p t i o n that G is countable. N o w we want to know if there exists, lity measures
among all invariant Radon probabi-
on T, a p r o b a b i l i t y measure ~ such that ~(T 0) = I, when
the countable group G is amenable.
Because T O is not compact, we cannot
apply the fixed-point p r o p e r t y to get the answer.
Since every Radon p r o b a b i l i t y measure on T is inner regular, we get
(T 0)
=
Sup ~ (K) KcT 0
w h e r e K runs in the set of all compact subspaces of T contained in T O .
R e v e r s i n g the order in which we take the least upper bounds, we get
Sup ~(T 0) ~eK(C(T),s,G)
=
Sup Sup u(K) KCT 0 ~eK(C(T),s,G)
But the indicator of a compact set is an upper semi-continuous
function,
and we can use the ergodic theorem 3.1.3 to get the least upper bound of all invariant p r o b a b i l i t y measures on such a compact set.
Next, we have to study the compact sets contained in T O .
8.1.5.
Definition.
Given an element • of TO, d r stands for the function
on GxG w i t h integer values defined by
d (g,h)
=
i +
=
0
I{keG, g~k~h or h~k~g}[
if
g
=
if
g # h
h
It is a distance on G.
8.1.6. Proposition.
Let d be a distance on G with integer values.
K d of all total orders such that d
The set
is less than or equal to d is compact.
122
Proof. equal
Consider the function dg,h on T with values to the n u m b e r
is
of elements between g and h plus one if this number
is finite and to += otherwise.
This function is lower semi-continuous.
The set of all total orders such that dg,h(T) d(g,h)
in N where dg,h(T)
is less than or equal
to
is then a closed subset of T.
The set K d is the intersection of all these closed sets when g and h run in G, and it is therefore compact.
8.1.7.
Theorem.
Let G be a countable amenable group.
Then,
there exists
a Borel probability, w h i c h is invariant under the right translations,
on
the set T O of all total orders on G that are isomorphic to the natural order of Z.
Proof.
It shall be sufficient to find,
a distance d on G with integer values
for every positive real number e, such that the following holds for
the c o r r e s p o n d i n g compact set Kd:
Sup ~ (K d) ueK(C(T),s,G)
>
I -
We then find an invariant Radon p r o b a b i l ~ y measure on T that gives the whole mass to the union of all K d. This union is p r e c i s e l y the set of all total orders for which there are only finitely m a n y elements b e t w e e n two given ones.
On the other hand,
thanks to an easy invariance argument,
the set of all
total orders with a first or a last element has a measure 0 for every invariant Radon p r o b a b i l i t y measure on T. The difference is the set T O of all orders that are isomorphic to the order of Z. Thus we must look for distances on G with integer values,
in particular
ones that have the following form:
d(g,h)
=
d(g) + d(h)
if
w h e r e d is a integer function on G equal
Take a converging sequence
(l-e i)
>
to 0 at e.
(~n) of p o s i t i v e real numbers
such that
i=l
g#h
1 -
E
less than i
123
and an increasing
sequence
(D n) of finite parts,
containing
the unit
element e of G, whose union is G. Because G is amenable,
there exists,
for every subscript
i, a finite
part A i of G such that
(IAil/IDi~Ail).(I
- mD!Ai)/IAil)
>
I - ~i
i
Define
Next,
therefore
the function d in the following way:
d(e) =
0
d(g) =
IAII - i
if
geDl\{e}
d(g) =
IAnl
if
geDn\Dn_ I
I
denote by K n the following
Kn
=
compact
subset of T:
{meT, VgeD n d (e,g)
Apply to the 2n+l-uple
(A,AI,..,An,DI,..,D n) the iterated
We then get an adapted
iterated
Now construct
the following
tiling
tiling
lemma.
(PI .... ,Pn ) of A.
order m on G:
-
let the restriction
of T to every atom of PI be arbitrary
-
get the restriction
to an atom of P2 by setting an arbitrary
order on the set of all atoms of PI contained taking the lexicographical -
in it, and then by
order on it
and so on in such a way that the atoms of all partitions intervals
for the built order on A
decide that all elements An easy computation
out of A are less than those of A.
then shows that the ergodic average
(III^I).geAI IKnoTg(m) is greater
Pi are
than
(i - m D
i=n ( A ) / I A I ). ~ (I - e i ) n i=l
124
Taking the limit along the ameaning filter for the part A, and applying the ergodic theorem 3.1.3, we get
Sup ~(K n) ueK(C(T) ,s,G) The compact set K d contained Applying now corollary
in T O is the intersection
of all K n.
of K d is greater
than or equal to i - e.
the proof of the theorem.
8.2. EQUIVALENCE
8.2.1.
i=n ~ (i - ~i ) i=l
1.3.7, we deduce that the least upper bound of all
invariant Radon probabilities This completes
>
OF COUNTABLE
GROUPS
Definitions.
Endowed with the product topology of discrete
topologies,
the set H G of
all mappings from the countable Polish space.
group G in the countable group H is a
Its subspace A(G,H),
of all elements
consisting
for which the image of
the unit element of G is the unit element of H, is a closed subspace, and thus a Polish space for the induced topology. Let g be an element of G. The right translation by g does not preserve the subspace A(G,H). We must now define a mapping T g from A(G,H) following
definition Tg(a)
to itself by giving it the
on the graphs: =
{(glg-l,hlh-l),
(gl,hl)ea}
where h is chosen in such a way that (e,e) belongs h is the image of g by a. Hence, using the functional Tg(a) (g ')
notation, =
to Tg(a),
i.e. that
T g is defined by
a(g'g) .a(g) -i
It is easy to verify that the mappings
T g are continuous
and that they
constitute
an action of G on the Polish space A(G,H) by homeomorphisms.
We denote,
too, by A(G,H)
the corresponding
dynamical
system.
125
The subset I(G,H)
consisting
of all one-to-one
elements
closed and invariant under the action of G. Once again, stands
for the corresponding
The set B(G,H) mappings
But,
B(G,H)
also denotes
(a G6 in a Polish
in this system,
possible
I(G,H)
is
also
system.
is the subspace of I(G,H)
from G onto H. This subspace
of G. Once more, system
dynamical
of A(G,H)
consisting
of the one-to-one
is also invariant under the corresponding
the action
standard
dynamical
space is a Polish space).
the two groups play symmetrical
to define an action of H on B(G,H)
roles and it is
in a way very similar
to
that of G. It is not obvious, bility measure
however,
that there exists
for this dynamical
an invariant
system when G is amenable.
dard space is not a compact one and it is not possible point property because
Borel probaThe stan-
to use the fixed-
the convex set of all Borel probability measures
is also not compact. The next proposition makes
clear the symmetry between
the roles of the
two groups when acting on the set B(G,H). 8.2.2.
Proposition.
Let B(G,H)
be the set of all one-to-one mappings
from G onto H, giving to the unit element of G the unit element
of H as
its image. Consider
the action T of G on this space given by Tg(x) (g ' )
=
x(g'g) .x(g) -I
and the action U of H on it defined by uh(x)(g ') Then,
=
triction of the product Moreover,
Proof.
where
g = x-l(h)
for the topology of a Polish space on B(G,H),
mappings under
x(g'g).x(g) -I
topology of discrete
ones,
obtained
as the res-
the T g are continuous
and the U h are Borel ones. every Borel probability measure
the action of G, is also invariant The continuity
on the space which is invariant under
that of H.
of the T g is straightforward.
In order to state the Borel m e a s u r a b i l i t y
of the transformation
U h, we
126
need only show that, elements
for every pair
This makes
obvious
O ({x,x(g)=h geG
that this set is a Borel
the invariance
for the cylinder A because
set because
it is a F
O
in the
=
.o. and X(gn)=h n}
sets generate
the Borel o-algebra.
U ({x(glg)x(g)-l=hl}~...~{X(gng)X(g)-l=hn}N{x(g)=h}) geG equality where
(uh)-I (A)
=
Using now the invariance
the union is disjoint:
0 (Tg) -i (A~{ x(g-l)=h-l} ) geG of ~ under T, we obtain
~((uh)-I(A))
8.2.3.
=
~ geG
=
,(A)
(Afl{x(g-l)=h-l})
Remark.
The mapping
from B(G,H)
ment of B(G,H)
its inverse
mapping
Nevertheless,
to the Borel
tions becomes
the standard structure,
is invariant
is invariant under
Remark.
the actions
space
fact,
giving
to every ele-
is not a continuous it is a Borel
B(G,H)
the postulated
and forget symmetry
one°
isomorphism. the topology
of the two ac-
clear.
is then an easy corollary
probability
to B(H,G),
as image,
and this is an important
if we consider
leading
this property
set, we get
the following
8.2.4.
and x(g'g)=h'.x(g)})
of ~ under U, we must only verify
{x(gl)=h I and
these cylinder
(uh)-I(A)
This
as follows:
sets
=
For such a Borel
Thus,
the set of all
space B(G,H).
To state
Hence
in GxH,
such that uh(x)(g ') = h' can be decomposed {xeB(G,H),uh(x)(g')=h '} =
Polish
(g',h')
under
of the previous
proposition
one of the two actions
that a Borel
if and only if it
the other one.
A probability
measure
on B(G,H)
T and U of G and H, is ergodic
which
is invariant
for one action
under
if and only if
127
it is ergodic
for the other.
Indeed,
the measurable
because
the orbits under the two actions
8.2.5.
Definition.
if there exists
invariant
sets are the same for the two actions
Two countable
groups G and H are said to be equivalent
a probability measure
on the Polish space B(G,H) which
is invariant under the two previously This relation
is clearly a reflexive
term "equivalence" 8.2.6.
Theorem.
is a Borel
by proving
the transitivity
tensor product
Proof.
one. We justify the
of this relation.
Let G, H and K be three countable
groups.
on B(G,H)
Suppose that
and v a Borel
invar-
on B(H,K).
the probability measure ~ v
actions
defined actions of the groups. and symmetrical
invariant probability measure
iant probability measure Then,
are the same.
of ~ and v under
on B(G,K) which
is the image of the
the composition
is invariant under
from the product
space B(G,H)xB(H,K)
the
of G and K on B(G,K). The composition mapping
B(G,K),
defined by (x,y) --> z = xoy,
Borel o-algebras,
In order to state the invariance of G on B(G,K),
is measurable
allowing us to define
the image ~ v
of the measure ~ v
it is sufficient,
with respect
to
to the
on B(G,K). under
as we have previously
the action V
noted,
to prove
the equality ~ v ( ( V g)-l(A))
when the Borel
A
We can write
=
set A is a cylinder
=
the following
equalities:
n
(vg)-l({z,z(gi)=ki})) 1
set
{z(gl)=kl}~... N{Z(gn)=kn}
n
~v(~
~v(A)
= ~v(~({z,z(gig)z(g)-l=ki})) 1 n = ~ ~v(({z,z(g)=k})~A({z,z(gig)=kik})) keK 1
128
Using the definition of the measure ~ v with the tensor product ,®v, the above number is equal to the following sum: n
~ ~ {~(({x,x(g)=h}) keK heH (hi)eH n
O N({x,x(gig)=hih})). 1 n (({y,y(h)=k}) ~ ~({y,y(hih)=kik}))} 1
Employing the actions T and U, we see that this sum is equal to n
~ ~ {~((Tg)-l(({x,x(g-l)=h-l}) keK heH (hi)eH n
0 ~({x,x(gi)=hi}))). I n ~ ~({y,y(hi)=ki})))} 1
((U h)-l(({y,y(h-l)=k-1}) Next u s i n g t h e i n v a r i a n c e we find that the number
of u u n d e r T and the i n v a r i a n c e
o f v u n d e r U,
n ~ v ( ~ ( v g ) -I ( { z , z ( g i ) = k i } ) ) 1 is equal to n
k~eK h~eH (hileH n {~(({x,x(g-l)=h-l}) A n({x,x(gi)=hi})).l n v (({y,y(h-l)=k -I}) ~ ~({y,y(hi)=k i}))} I The following equality then results from the o-additivity product: n
~ v ((vg) -I (N({ z ,z (gi) =ki} ) ) I
of the tensor
n
=
~ v (N({ z,z(gi)=ki} ) ) I
and the "convolution product" is then invariant under the action V of G on B(G,K). 8.2.7. Proposition. All countable amenable groups are equivalent sense of definition 8.2.5.
in the
Proof. Since the involved relation is an equivalence one, we need only state the equivalence of any amenable group with a fixed one, Z. The measurable space B(G,Z) can be imbedded in the compact space of all total orders on G by identifying an element x of B(G,Z) with the total order on G, which is the image by x of the natural order of Z.
129
This embedding
is clearly a one-to-one mapping
space T O of T. In addition, two measurable B(G,Z)
this mapping
from B(G,Z)
is an isomorphism between
spaces and gives a conjugacy between
and the action of G by right translations
The existence results
of an invariant
from the existence
onto the subthe
the action of G on
on T O .
Borel probability measure
on B(G,Z)
then
of a similar measure on TO, which we proved
in section 8.1. 8.2.8.
Proposition.
an invariant
Given two equivalent
countable groups G and H, and
Borel probability measure ~ on B(H,G),
for every element m of the cone E(G),
we can construct,
a real function m
on the set F(H)
of all finite parts of H by setting m (A)
f
=
m(x(A))
d~(x)
B(H,G) This function m Proof.
belongs
to the cone Z(H)
and we get q(m ) ~ q(m).
These results are easily obtained by integration,
invariance 8.2.9.
follows
from the invariance
Proposition.
Given two equivalent
countable
groups G and H, and
an element m of the cone E (G), denote by i e the upper function on the compact
set T(G)
and the right
of ~.
semi-continuous
of all total orders on G defined
example
3.1.13;
denote by Je the upper
semi-continuous
defined
in the same way from the invariant
in
function on T(H)
capacity m , built
from m
with the invariant measure ~ on B(H,G). Let ~H be a Radon probability measure
on T(H) which is invariant under
the action of H by right translations. Consider
the measurable mapping
where x(T)
(x,r) --> x(r)
from B(H,G)xT(H)
to T(G)
is defined by (a,b)e~
This mapping
-~ 3
(x(a),x(b))ex(T)
sends the tensor product ~®~H to a probability measure ~G
on T(G). Then, Proof.
the integral
of Je for ~H is equal
The following
ie(~) fT(G)
equality results
d~G(T)
=
f
to the integral
of i e for ~G.
from the very definition of ~G:
i (x(T)) B (H,G)xT (H) e
d~ (x) d~H(~)
130
The integral
for p of ie(X(~))
is equal
to je(~)
and the result is
obtained. 8.2.10.
Corollary.
A countable
one is also amenable. amenable
group,
The mean value of an invariant
and the mean value of the invariant
it on another equivalent Indeed,
group which is equivalent
group,
if H is an amenable
to an amenable
capacity
on an
capacity built from
are equal.
group and,
if G is equivalent
=
=
to H, we get,
for every element m of t(G),
q(m)
<
=
q(m )
This shows first that q(m)
~H(Je )
is equal
invariant probability measures according
to example
3.1.13.
to the least upper bound of all
of ie, and,
The equality
8.3. ROKHLIN'S LEMMA AND HYPERFINITENESS
8.3.1. space
First, (X,~,~)
recall Rokhlin's
lemma.
is said to be aperiodic
from 0, the measure
~G(ie)
then that G is amenable q(m) = q(m ) results.
OF COUNTABLE AMENABLE
A transformation
GROUPS
T of a probability
if, for every integer n different
of the set of all points
invariant under T n is equal
to 0. If the standard dynamical says that,
system
for every positive
n, there exists a measurable
(X,~,~ ,T) is aperiodic,
is greater
disjoint and the measure
of their
than 1 - E.
We cite here the strongest result was achieved by Connes, If R is a countable Lebesgue
lemma
integer
set F in X such that the images Ti(F), where
i runs between 0 and n, are m u t u a l l y union
Rokhlin's
real number E, and every positive
space,
in the field of hyperfiniteness,
amenable measurable
this equivalence
single transformation; of the probability
which
Feldman and Weiss:
this means
equivalence
relation
on a
relation can be generated by a that there exists an automorphism
space such that the graph of R is the set of all
(x,Tn(x)) where x runs in X and n in Z, up to null sets obviously.
T
131
We shall restrict ourselves in the proof of this result to the case of the action of an amenable group on a p r o b a b i l i t y space w i t h an invariant p r o b a b i l i t y measure.
8.3.2. Definition.
A standard dynamical
system is a dynamical
isomorphic to an action by Borelian automorphisms preserves
system
on a Polish space that
a Borel p r o b a b i l i t y measure.
The following measurable
section theorem is useful
to the proof of an
analog of Rokhlin'lemma.
8.3.3. Measurable section theorem. Let X be a Polish space and R be an e q u i v a l e n c e relation on X. Suppose that all equivalence classes are closed and that the saturated set of every open set is m e a s u r a b l e with respect to a given o - a l g e b r a Then,
that contains all open sets.
there exists a m e a s u r a b l e part of this o - a l g e b r a
that meets every
e q u i v a l e n c e class in one and a single point.
The proof can be found,
8.3.4.
Definition.
(X,~,~)
for instance,
o b v i o u s l y the unit one),
g
(I).
An action T of a group G on a p r o b a b i l i t y space
is said to be aperiodic
I
in Bourbaki
= {xeX,
if, for every element g of G (except
the m e a s u r e of the m e a s u r a b l e
set
Tg(x) = x}
is equal to 0. 8.3.5.
Definition.
Given an aperiodic action of a countable group on a
standard probability space
(X,~,~), denote by R the equivalence r e l a t i o n
whose graph is a m e a s u r a b l e part of XxX
(x,y)eR
<
3
~geG,
Tg(x) = y
8.3.6. Definitions. Under the previous hypotheses,
a tower is a symmetrical and transitive
m e a s u r a b l e r e l a t i o n on X whose graph is contained in R, and whose every class
is finite. The basis of the tower t is then the m e a s u r a b l e
of all points of X that verify
(x,x)et.
set t
132
Thus,
the restriction
Denote
then by t(x)
relation
of a tower to its basis is an equivalence
relation.
the class of a point x of ! for the equivalence
t.
We shall make no distinction between and shall only consider non-trivial not a null
towers towers,
that agree almost everywhere i.e. towers whose basis is
set.
Our aim is now to build towers with a basis as large as possible, such that the forms of G corresponding
to the classes
and
are as invariant
as possible. 8.3.7. Lemma dynamical
(separation
system.
Proof.
there exists
We need only repeat
several
part A' of A of positive
of F are m u t u a l l y disjoint.
times the following (f,f')
argument with all
is any pair of different
ele-
of F:
fixed point,
transformation
it is possible
tive measure,
last result
to a theorem,
an aperiodic
in A,
Let
to its image
of T to A is conjugate with the identity.
(I), such a transformation
Theorem.
included
are disjoint.
the proof of which can be found for instance
then almost everywhere 8.3.8.
any
for every event A with a posi-
part of the standard space A is equal
by T, the restriction Billingsley
to find,
space without
can be proved by contraposition.
If every measurable Thanks
of a standard
another event A' with a positive measure
such that A' and T(A')
and
,u,T) be a standard aperiodic
a measurable
T fo(Tf') -I, where
If T is a measurable
This
(X,
translates by the elements
transformations ments
Let
For every finite part F of G and every event A with a
positive measure, measure whose
lemma).
is isomorphic
equal to it, which contradicts
(X,~,u,T)
be a standard dynamical
action of a countable
amenable
in
to the identity and the aperiodicity. system where T is
group G by automorphisms;
let F be a finite part of G and A be a measurable
part of X with a
positive measure. For every positive contained
real number e, there exists
in A and which verifies
I
the following
a tower t whose basis inequality:
{u({xet,
Tf(x)eA and
(x,Tf(x))~t})
+
~({xet,
Tf(x)eA and
(Tf(x),x)~t})
}
feF <
E .~ (t)
is
133
Proof. Consider, for every non-empty finite part A of G, the following measurable function whose integral is equal to the measure of A: (I/{A{). I IAoTg geA For every element f of F, the integral of the function (IIIAI). IAoTg. iAoTfg geA, fg~A is coarsely bounded above by mF(A)/IAI . Thereupon follows {{xeAy,Tf(x)eA and Tf(x)~Ay}{
d~(y)
X <
(mF(A)/IAI).(I/~(A)). f IAyqA{ X
d~(y)
and the analogous result f l{x~Ay,xeA,Tf(x)eAY ~A}I d~(y) X <
(mF(A)/IAJ).(I/~(A)). ~ {AyNA{ X
dr(y)
Choose the finite part A such that mF(A)/{AI is less than ~.~ (A)/21 FI . The measurable part B of all y in X such that the sum {{xeAyAA,Tf(x)eA and Tf(x)~Ay}[
+ {{x~Ay,xeA,Tf(x)eAnAy}I
is less than e.IAy~AI , therefore has a strictly positive measure. Using the separation lemma, it is possible to find a measurable part C of B with a strictly positive measure and whose translates by all elements of A are disjoint. The tower t is then defined in the following way. Its basis is the set (A.C)~A, and two points in this basis are related if the single element of C of which they are translates by an element of A is the same. This tower answers the question. 8.3.9. Corollary. If the basis of the tower t has not a measure equal to i, and if t verifies the inequality X ~({xet,Tf(x) et and (x,Tf(x))~t}) feF
__< E.~ (t)
134 then there exists a tower t', whose basis
is disjoint
the tower t", union of t and t', still verifies Proof.
from t such that
the previous
inequality.
We need only apply theorem 8.3.7 to the complementary
part of !.
The computation
is elementary.
8.3.10 Theorem.
For every finite part F of G and every positive
number e, there exists
real
a tower t such that the sum
u ({x,(x,Tf(x))~t}) feF is less than E. This means
that it is possible
to find towers
that almost
contain the graphs of the T f. Proof.
Consider
towers
:
the following
strong order relation on the set of all
t I is less than t 2 if there exists
t 3 such that t 2 is the
disjoint union of t I and t 3. The set of all towers u({xet,
that verify Tf(x)et and
(x,Tf(x))~t})
<
E.u(t)
feF is inductive hand,
for the previous
order.
Corollary
8.3.8 shows,
that this set is not empty and, on the other hand,
element has X for its basis extendable. 8.3.11.
Such a maximal
Corollary.
on the one
that a maximal
since a tower with another basis is strictly element verifies
There exists
the predicted
an increasing
sequence
inequality.
of towers whose
union is R. Proof.
Let Fn be an increasing
sequence
is G, and let (E n) be a converging For every positive
of finite parts of G whose union
sequence of positive
integer n, the previous
real numbers.
theorem gives a tower tn that
verifies
U ({x' (x'Tf (x))~tn }) feF Denote greater
~
~n
n
then by t' the tower which is the intersection n than
or
equal
to
n.
of all t for m m
135
We get the following
f The
sequence
elements
inequality:
e[ ~({x, F n of
all
contains
t'
(x,Tf(x))~tn })
n
the
is
<
[ Sm
=
n
an increasing
graphs
of
all
one
Tf,
and
where
the
f runs
union in
of
G, u p
all to
its null
sets. 8.3.12.
Corollary:
Let (X,~,u,T)
the Rokhlin's
be a standard aperiodic
action of an amenable Then,
lemma for countable
countable
for every triple
n two positive
dynamical
amenable
groups.
system where T is an
group G.
(D,~,n), where D is a finite part of G, and ~ and
real numbers,
there exists a finite set I of pairs
(Ai,Ai),
with the following properties: I)
for every i in I, A i belongs
2)
the images Tg(Ai) of the Ai, where i is in I and g in Ai, are disjoint (up to null sets) and the measure of their union is greater
to M(D,~)
and A i to the o-algebra
than 1 - n.
Proof. Let L be a maximal
tower such that the sum
u ({x, (x,Td(x))~t}) deD is less than 6.n. The equivalence
relation
section theorem.
t verifies
all hypotheses
Let A be a measurable
disjoint union of all measurable
section.
of the measurable
Set A is the countable
sets AA, where A is a finite part of G,
that are defined by AA The previous
=
{xeA,
integral
AeFIG)fA
On the other hand, I
=
(x,y)et <
9
geA, y=Tg(x)}
is then equal to
(d[eDmd(A))
du(x)
the following I [A[ .~ (AA) AeF(G)
decomposition
holds:
136
Hence the inequality
AeF(G)
mD(A).V (AA)
<
6.~. AeF(G)
IAJ .~ (AA~
and the upper bound
[
A ~M(D,~ ) Then,
there exists a finite
M(D,~)
such that the finite
This achieves 8.3.13.
the aperiodic
Let
everywhere
n
set of finite parts of G that all belong sum of all IAJ .V (AA)
Let
(X,~,v,T)
is greater
be a standard dynamical
action of a countable amenable a bimeasurable
transformation
orbits almost everywhere
Proof.
<
to
than 1 - n.
the proof of the predicted result.
Theorem.
There exists whose
fAr.~(A A)
group G.
S of the probability
space
agree with those of T.
(t n) be an increasing equal
system where T is
sequence of towers whose union is almost
to R.
Apply the measurable
section theorem
to the equivalence
relation t n on
its basis. We then get a measurable
set A n which meets every equivalence
class in
a single point. Because
the sequence of relations
is increasing,
the saturated part of
A n for tn+ 1 is equal to the basis of tn+ I. Applying the measurable section theorem to the restriction the part An, we can find An+ 1 contained With this decreasing
sequence
of tn+ 1 to
in A n .
of measurable
sections,
we can construct
some order relations. First fix a reference stands
total order on G. For n greater
than I, An(X)
for the single finite part of G such that the equivalence
of x for the restriction Denote by Ak(X)
of tn on An_l,
tn(X)NAn_l,
class
is An.(X).
the single element of A k related by tk with x, and write
=
TgI(A l(x))
with gleAl(Al(X))
Al(X)
=
Tg2(A2(x))
with g2eA2(A2(x))
An_l(X)
=
Tgn(An(X))
with gneAn(An(X))
137
The order T n is defined on the basis related by tn for the lexicographical Two points
t of t by comparing -n n order.
two points
related by tn have the same image by A n and are characterized
by the n-uples (gn,...,gl). Thus, we never put any new element between
two elements
once they have
been related. In the limit, we obtain a measurable which
order relation T on the orbits
is a total one in the sense that two equivalent
points
are com-
pared by the order. For almost every x in X, there are only finitely many points between x and Tf(x). In order to achieve transformation 8.3.14.
Definition.
countable
the proof of the theorem,
Two aperiodic
actions
(X,
groups are said to be orbitally
measurable
consider
S that sends a point to its successor
,~,G) and
equivalent
invariant part X' of X with a measure
iant part Y' of Y with a measure
the measurable
for the order ~. (Y,
,v,H) of two
if there exist a
i, a measurable
i, and a bimesurable
invar-
mapping between X'
and Y' such that the image of the orbit of every point
is the orbit of
its image. 8.3.15.
Remark.
phism between
The isomorphism the orbital
An equivalence
the o-algebras
in the previous
of invariant
of these o-algebras
sense gives an isomor-
events.
is then a necessary
condition for
equivalence.
Let's give Dye's result
in the ergodic
case, where
these o-algebras
are
trivial. 8.3.16. valent
Theorem.
Two ergodic actions
in the sense of definition
of Z on Lebesgue
The proof can be found in a paper by Dye 8.3.17.
Corollary.
Indeed,
hyperfiniteness
is amenable
ergodic
is transitive.
systems where the coun-
are equivalent.
gives the equivalence
standard action of Z, and Dye's since equivalence
(i).
All standard aperiodic
table group of transformations
spaces are equi-
8.3.14.
of such a system with a
theorem allows us to conclude
the proof
138
8.3.18.
Remark.
amenable
Conversely, allows
It is simple enough to show the hyperfiniteness
group action on a standard the existence
space from Rokhlin's
of a single transforamtion
to go from the usual Rokhlin's
for amenable groups.
It is therefore
invariant
described
capacities
of an
lemma.
with the same orbits
lemma to the version that we gave sufficient
to use the transfer
of
in section 8.2.
8.4. REFERENCES
For the results As regards
present author's Our note
about almost tiling,
tiling particular
consult
joint work with D. Pinchon
(4) concerns
amenability
The proof of the hyperfiniteness relation was done by Connes, Rokhlin'
refer to Ornstein and Weiss
groups,
of groups
for example,
that are equivalent
Feldman and Weiss
ergodic theory because of its fundamental to Conze
(5).
(4) and the
(3).
of an amenable
lemma is shown in his exposition
tions; refer,
their paper
countable
to Z.
equivalence
(I).
paper
(I) and in many books on
character
for so many applica-
(i).
A precise reference
for the measurable
Bourbaki on general
topology,
chapter
section theorem is the work of 9, page TG IX 70 of the last
edition. Dye's article
(I) gives different
ness and the orbital
equivalence
equivalent
The comparison between isomorphism (I, pp. 66-73).
definitions
of ergodic systems and conjugacy
of hyperfinite-
is stated there.
is made by Billingsley
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INDEX
abstract dynamical systems adapted 5.2.9 almost subadditive
2.1.5
4.4.3
almost subadditive ergodic theorem ameaning filter
4.4.6
2.2.8
amenable group 2.2.1 aperiodic group action
8.3.4
asymptotic -algebra 7.3.1 attractive interaction 6.2.13 automorphism of a probability space B
Bernoulli scheme
2.1.8
Birkhoff's theorem cocycle
2.1.4
3.3
6.1.2
conditionnal entropy 4.2.10 conjugacy 1.4.7, 4.1.4 covariant
4.4.4
Dini's lemma 1.3.6 dynamical Boolean algebra entropic distance
4.1.2
4.3.6
entropy of a dynamical system entropy of a partition
entropy of probabilty vectors equilibrium measures 1.4.1
finite modification fixed point property
4.2.1
7.2.5
equivalence of countable groups extremal point
4.3.2
4.2.7
6.2.1 2.1.11
8.2.5
146
generator
4.3.5
Gibbs measures Hahn-Banach Holley'
6.1.5
theorem
relation
information
1.1.4
6.4.5
relative
invariant
cocycle
invariant
interaction
Invariant
local specification
invariant
7.1.7
2.1.7
Ising model isomorphism
6.2.11
of two abstract theorem
Kolmogorov-Sinal local energies
dynamical
4.3.14
7.2.4 6.1.3
mean entropy of a partition mean entropy of order
capacity
section theorem
mixing properties
8.3.3
3.2
open segment property orbitally equivalent ordered partition
2.2.13
7.3
norm ergodic theorems
1.4.2 dynamical
systems
4.3.7
isomorphism
theorem
4.3.16
5.2.10
pair interaction partition
4.3.1
5.3.3
mean value of an invariant
overlap ratio
systems
6.3.3
theorem
local specification
Ornstein's
2.2.2
6.3.5
Kirkwood-Salsburg
N
theorem
6.2.10
Ising interaction
measurable
7.1.1
version of Hahn-Banach
interaction
4.4.1
7.1.15
invariant measure
M
to a partition
6.2.12
4.2.3
phase transition
6.3.1
pressure of a continuous
function
5.2.3
8.3.14
4.
147
pyramidal decomposition
2.2.15
quasi-invariant measures Radon measures
6.2.3
1.3.1
Riesz representation theorem
1.3.3
Rokhlin's lemma 8.3.1 Rokhlin's lemma for countable amenable groups saddle ergodic theorem
3.4
separated 5.2.1 Shannon-McMillan theorem
4.4.2
standard dynamical system strongly of order
8.3.2
5.3.3
strongly mixing 7.3.4 strongly subadditive functions subadditive functions sublinear function
2.2.10
3.1.5
I.I.I
supermodular interactions
6.4.2
symbolic dynamical systems T
2.1.2
tiling sets 8.1.1 tiling lemma 8.1.2, 8.1.3 topological dynamical systems
2.1.1
topological entropy 5.1.1 topologically transitive dynamical systems total orders tower
3.2.9
2.1.3, 8.1.4
8.3.6
uniquely ergodic topological dynamical systems variation table
2.2.18
variational principle W
8.3.12
weak mixing
7.3.5
5.1.3
3.2.8
