Ergodic Theory and its Applicationsijaema.com/gallery/95-january-3219.pdfergodic theory. Key Words...
Transcript of Ergodic Theory and its Applicationsijaema.com/gallery/95-january-3219.pdfergodic theory. Key Words...
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Ergodic Theory and its Applications
Christy Tom Mathews, Mily Roy
Department of Mathematics,Deva Matha College Kuravilangad,Kerala,India
Abstract
This paper provides an indepth knowledge about ergodicity,its definition,examples and certain
characterization theorems.It discusses about the major ergodic theorems and the invariant
measures for continous transformations.It also deals with the number theoretic applications of
ergodic theory.
Key Words
Ergodicity,transformations,mixing,weak mixing,strong mixing,invariant measures,normal numbers
INTRODUCTION
In its broad interpretation ergodic theory is the study of the qualitative properties of actions of groups on
spaces. The space has some structure,that is, the space may be a measurable space, or a topological space,
or a smooth manifold and each element of the group acts as a transformation on the space and preserves the
given structure(for example, each element acts as a measure-preserving transformation, or a continuous
transformation, or a smooth transformation).
The word โergodicโ was introduced by Boltzmann to describe a hypothesis about the action of
๐๐ก ๐ก โ ๐ on an energy surface ๐ปโ1๐ when the Hamiltonian ๐ป is of the type that arises in statistical
mechanics. Boltzmann had hoped that each orbit ๐๐ก ๐ฅ ๐ก โ ๐ would equal the whole energy surface
๐ปโ1๐ and he called this statement the ergodic hypothesis. The word โergodicโ is an amalgamation of the
Greek words ergon (work) and odos (path). Boltzmann made the hypothesis in order to deduce the equality
of the time means and phase means which is a fundamental algorithm in statistical mechanics. The ergodic
hypothesis, as stated above is false. The property the flow needs to satisfy in order to equate time means and
phase means of real-valued functions is what now called ergodicity.
It is common to use the name ergodic theory to describe only the qualitative study of actions of
groups on measure spaces. The actions on topological spaces and manifolds are often called topological
dynamics and differentiable dynamics. This measure theoretic study began in the early 1930โs and the
ergodic theorems of Bikchoff and von Neumann were proved then. During recent years ergodic theory had
been used to give important results in many branches of mathematics.
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2. ERGODICITY
2.1 DEFINITION AND CHARACTERISATION
Definition 2.1:
Let (X, โฌ, m) be a probability space. A measure-preserving transformation T of (X, โฌ, m) is called ergodic
if the only members B of โฌ with Tโ1B = B satisfy m B = 0 or m B = 1.
Theorem 2.1.1:
If T: X โ X is a measure-preserving transformation of the probability space X, โฌ, m then the following
statements are equivalent:
(i) Tis erogodic.
(ii) The only members B ofโฌ with m Tโ1B โ B = 0 are those with m B = 0 or m B = 1.
(iii) For every A โ โฌ with m(A) > 0 we have m โชn=1โ TโnA = 1.
(iv) For every A, B โ โฌ with m(A) > 0, m(B) > 0 there exists n > 0 with
m(TโnA โฉ B) > 0.
Theorem 2.1.2 :
If (X, โฌ, m) is a probability space and T: X โ Xis measure preserving then the following
statements are equivalent :
(i) Tis ergodic.
(ii) Whenever f is measurable and f โ T x = f x โx โ X then f is constant a.e.
(iii) Whenever f is measurable and f โ T x = f x a.e. then f is constant a.e.
(iv) Whenever f โ L2(m) and f โ T x = f x โx โ X then f is constant a.e.
(v) Whenever f โ L2(m) and f โ T x = f x a.e. then f is constant a.e.
Proof:
Trivially we have (iii)โ(ii),(ii)โ(iv),(v)โ(iv),and(iii)โ(v).So it remains to show (i)โ(iii)and(iv)โ(i).
Claim:(i)โ(iii).
Let Tbe ergodic and suppose f is measurable and f โ T = f a.e. We can assume that f is real-
valued for if f is complex-valued we can consider the real and imaginary parts separately. Define
,fork โ Z and n > 0,
X k, n = x k 2n โค f x <(k + 1)
2n
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= fโ1 k 2n ,(k + 1)
2n .
We have
Tโ1 X k, n โX k, n โ x f โ T x โ f x
And hence
m Tโ1 X k, n โX k, n = 0
So that by (ii) of theorem2.1.1, m X k, n = 0or 1.
For each fixed n โชkฯตZ X k, n = X is a disjoint union and so there exists a unique kn with
m X kn , n = 1. Let Y =โฉn=1โ X kn , n .Then m Y = 1 and f is constant on Y so that f is
constant a.e.
(iv)โ(i):Suppose Tโ1E = E, E โ โฌ. Then ๐ณEฯตL2 m and
๐ณE โ T x = ๐ณE x โx โ X so, by (iv) ,๐ณE is constant a.e. Hence ๐ณE = 0 a.e.or ๐ณE = 1 a.e. and
m E = โซ๐ณE dm = 0 or 1.
Remarks 2.1.1:
1. A similar characterisation in terms of Lp m functions (for any p โฅ 1) , is true, since in the
last part of the proof ๐ณE is in Lp m as well as L2 m .
2. Another characterisation of ergodicity of T is : whenever f: X โ R is measurable and
f(Tx) โฅ f x a.e. then f is constant a.e. This is a stronger statement than (iii).
The next result is useful to analyse which of the examples are ergodic and it also relates the idea
of measurable theoretic dense orbits (theorem 2.1.1 (iii)&(iv)) to the usual notion of dense orbits
for continuous maps.
We have the following theorem concerning rotations of the unit circle K.
Theorem 2.1.3:
The rotation T z = az of the unit circle K is ergodic(relative to Haar measure m ) iff a is not a
root of unity.
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Proof:
Suppose a is root of unity, then ap = 1 for p โ 0.Let f z = zp .Then f โ T = f and f is not
constant a.e. Therefore T is not ergodic by theorem 2.1.2 (ii).Conversely, suppose a is not a root
a root of unity and f โ T = f , fฯตL2(m) .Let f z = bnznโ
n=โโ be its Fourier series .Then
f az = bn
โ
n=โโ
anzn
and hence bn an โ 1 = 0 for each n.If n โ 0 then bn = 0, and so f is constant a.e. Theorem
2.1.2(v) gives that T is ergodic.
Theorem 2.1.4:
Let G be a compact group and T x = ax ,a rotation of G .Then T is ergodic iff {an}โโโ of G. In
particular ,if T is ergodic, then G is abelian.
Proof:
Suppose T is ergodic .Let H denote the the closure of the subgroup {an}โโโ of G .If H โ G then
by the result โif H is a closed subgroup of G and H โ G there exists a ฮณฯตG ,ฮณ โข 1 such thatฮณ h =
1โh โ Hโthere existsฮณฯตG ,ฮณ โข 1 but ฮณ h = 1โh โ H. Then
ฮณ Tx = ฮณ ax = ฮณ a ฮณ x = ฮณ x , and this contradicts ergodicity of T. Therefore
H = G.Conversely, suppose {an}nโZ is dense in G. This implies G is abelian. Let fฯตL2(m)
and f โ T = f.By the result โ If G is compact,the members of G form an orthonormal basis for
L2(m) where m is a Haar measureโ f can be represented as a Fourier series biฮณii ,where ฮณi โ G .
Then
biฮณi a ฮณii
x = biฮณii
(x)
so that if bi โ 0 then ฮณi a = 1 and, since ฮณi an = ( ฮณi a )
n = 1, ฮณi โก 1. Therefore only the
constant term of the Fourier series of f can be non- zero, i.e, f is constant a.e. theorem 2.1.2(v)
gives that T is ergodic.
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Theorem 2.1.5:
The two-sided (p0, p1, โฆ , pkโ1) shift is ergodic .
Proof:
Let ๐ denote the algebra of all finite unions of measurable rectangles. Suppose Tโ1E = E, E โโฌ. Let ฮต > 0 be given, and choose A โ ๐with m E โ A < ๐. Then
m E โ m(A) = m E โฉ A + m(E A ) โ m A โฉ E โ m(A
E )
< ๐(E A ) + m(A
E ) < ๐
Choose n0 so large that B = Tโ n0 A depends upon different coordinates from A. Then
m B โฉ A = m B m A = m(A)2 because m is a product measure . We have
m E โ B = m Tโn Eโ TโnA = m(E โ A) < ๐
and since E โ A โฉ B โ E โ A โช E โ B we have m E โ (A โฉ B) < 2๐
Hence m E โ m(A โฉ B) < 2๐
and m E โ m(E)2 โค m E โ m(A โฉ B) + m A โฉ B โ m(E)2
< 2๐ + m(A)2 โ m(E)2
โค 2ฮต + m A m A โ m E + m E m A โ m E
< 4๐,since ฮต is arbitrary m E = m(E)2 which implies m E = 0 or 1.
Theorem2.1.6:
If T is the p, P Markov shift (either one-sided or two sided) then T is ergodic iff the matrix P is
irreducible (i.e. โi, j โn > 0 withpij
(n)> 0 wherep
ij
(n)is the i, j -th entry of the matrix Pn).
Note:
Ergodic decomposition of a given transformation can be formulated as follows:
Ergodic transformations are the โirreducibleโ measure-preserving transformations and we would
like every measure โpreserving transformation to be built of ergodic ones. To get some idea how
this ergodic decomposition of a given transformation may be formulated consider a map T of a
cylinder X = 0,1 ร K given by T x, z = (x, az) where a โ K is not a root of unity. In other words
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T is the direct product of the identity map,I, of 0,1 and the rotation Sz = az of K. So T maps
each circular vertical section of the cylinder to itself and acts on each section by S.Now 0,1 ร K
can be partitioned into the sets x ร K and we can consider the direct product measure
m1 ร m2, where m1 is the Lebesgue measure on 0,1 and m2 is the Haar measure on K, as being
decomposed into copies of m2 on each element x ร K of the partition.
In other words we have a partition ฮถ of X on each element of which m1 ร m2
induces a probability measure and T induces a transformation which is ergodic relative to the
induced measure. This is the ergodic decomposition of T.It turns out that this procedure can
always be performed when (X, โฌ, m) is a nice measure space, namely a Lebesgue space.
Now if X is a complete separable space, m is a probability measure on the ฯ-
algebra โฌ(X) of Borel subsets of X and โฌ is the completion of โฌ(X) by m then (X, โฌ, m) is a
Lebesgue space. Suppose T is a measureโpreserving transformation of a Lebesgue
space (X, โฌ, m). Let โ(T) be the ฯ-algebra consisting of all measurable sets B with Tโ1B = B. The
theory of Lebesgue spaces determines a partition ฮถ of (X, โฌ, m) such that if โฌ(ฮถ) denotes the
smallest ฯ-algebra consisting of all members of ฮถ thenโฌ ฮถ = โ(T) in the sense that each
element of one ฯ-algebra differs only by a null set from an element of the other . Moreover
TC โ C for each element C of ฮถ . Also, the measure m can be decomposed into probability
measures mc on the elements C of ฮถ . The transformationT
C turns out to be ergodic relative to
the measure mc .This transformation is essentially unique .
Example of Ergodic Transformation:
Irrational rotations
Consider 0,1 , โฌ, ๐ , where โฌ is the Lebesgue ๐-algebra, and ๐ Lebesgue measure. For
๐ โ 0,1 ,consider the transformation ๐๐ : 0,1 โ 0,1 defined by ๐๐๐ฅ = ๐ฅ + ๐ ๐๐๐1 .
๐๐ is measure-preserving with respect to ๐ . If ๐ is rational then ๐๐ is not ergodic. Consider an
example, let ๐ = 1 4 , then the set
0 x 1
K
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๐ด = 0, 1 8 โช 1
4 ,3
8 โช 1
2 ,5
8 โช 3
4 ,7
8
is๐๐ โ invariant but ๐ ๐ด =1
2 .
Claim: ๐๐ is ergodic iff ๐ is irrational.
Proof of Claim:
Suppose ๐ is irrational,and let ๐ โ ๐ฟ2(๐, โฌ, ๐) be ๐๐ โ invariant . ๐can be represented in Fourier
series as
๐ โ ๐๐๐2๐๐๐๐ฅ
๐โ๐.
Since ๐ ๐๐๐ฅ = ๐ ๐ฅ , then
๐ ๐๐๐ฅ = ๐๐๐2๐๐๐ (๐ฅ+๐)
๐โ๐
= ๐๐๐2๐๐๐๐ฅ
๐โ๐๐2๐๐๐๐
= ๐ ๐ฅ
= ๐๐๐2๐๐๐๐ฅ
๐โ๐
Hence, ๐๐(1 โ ๐2๐๐๐๐ )๐2๐๐๐๐ฅ๐โ๐ = 0. By the uniqueness of the Fourier coefficients, we have
๐๐ 1 โ ๐2๐๐๐๐ = 0 โ๐ โ ๐. If ๐ โ 0, since ๐ is irrational we have 1 โ ๐2๐๐๐๐ โ 0. Thus,
๐๐ = 0 โ๐ โ 0,and therefore ๐ ๐ฅ = ๐0 is a constant. Then ๐๐ is ergodic.
2.2 THE ERGODIC THEOREM
The first major result in ergodic theory was proved in 1931 by G .D . Birkhoff. We shall state it for
measure-preserving transformation of a ฯ- finite measure space. A ฯ- finite measure on a
measurable space (X, โฌ) is a map m: โฌ โ R+โ โ such that m โ = 0, m โชn=1โ Bn =
m(Bn)โn=1 whenever Bn is a sequence of members of โฌ which are pairwise disjoint subsets of
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X, and there is a countable collection of {An}1โ of elements of โฌ with m An < โ โ๐ and
โชn=1โ An = X. The Lebesgue measure on (R
n, โฌ(Rn)) provides an example of a ฯ- finite measure.
Of course any probability measure is ฯ- finite .
Theorem 2.2.1 (BirchoffErgodic Theorem):
Suppose T: (X, โฌ, m) โ (X, โฌ, m) is measure-preserving(where we allow X, โฌ, m to be ฯ- finite )
and f โ L1(m).Then (1 n ) f(Ti(x))nโ1i=0 converges a.e. to a function f
โ โ L1(m). Also fโ โ T = fโ
a.e. and if m X < โ, then
fโdm = f dm.
Remark 2.2.1:
If T is ergodic then fโ is constant a.e. and so if m x < โ fโ = 1 m X โซ f dm a.e. If
X, โฌ, m is a probability space and T is ergodic we have โf โ L1 m
limnโโ
1 n f Ti x
nโ1
i=0
= f dm a. e.
Corollary 2.2.1(๐๐ฉErgodic Theorem of Von Neumann):
Let 1โค p < โ and let T be a measure-preserving transformation of the probability space
(X, โฌ, m) . If f โ Lp m there exists fโ โ Lp(m) with fโ โ T = fโ a.e.
1 n f Ti x
nโ1
i=0
โ fโ(x)
p
โ 0.
Corollary 2.2.2:
Let (X, โฌ, m) be a probability space and let T: X โ X be a measure-preserving transformation.
Then T is ergodic iff โ๐ด, ๐ต โ โฌ
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1 n m TโiA โฉ B
nโ1
i=0
โ m A m B .
Proof:
Suppose T is ergodic. Putting ๐ = ๐๐ด in theorem 2.2.1 gives
1 n ๐๐ด
๐โ1
๐=0
๐๐ ๐ฅ โ m A a. e.
Multiplying by ๐๐ต gives
1 n ๐๐ด
๐โ1
๐=0
๐๐ ๐ฅ ๐๐ต โ m A ๐๐ต .
and the dominated convergence theorem implies
1 n m TโiA โฉ B โ m A m B
nโ1
i=0
.
Conversely, suppose the convergence property holds. Let Tโ1E = E, E โ โฌ. Put ๐ด = ๐ต = ๐ธ in the
convergence property to get
1 n m(E)nโ1i=0 โ m(E)
2.
Hence m E = m(E)2 and m E = 0 or 1 .
Theorem 2.2.2 (Maximal Ergodic Theorem):
Let ๐: ๐ฟ๐ 1 ๐ โ ๐ฟ๐
1 ๐ be a positive linear operator with ๐ โค 1. Let ๐ > 0 be an integer and
let ๐ โ ๐ฟ๐ 1 ๐ . Define ๐0 = 0, ๐๐ = ๐ + ๐๐ + ๐
2๐ + โฏ + ๐๐โ1๐ for ๐ โฅ 1, and
๐น๐ = max0โค๐โค๐ ๐๐ โฅ 0 . Then โซ ๐ฅ ๐น๐ ๐ฅ >0 ๐๐๐ โฅ 0.
3. INVARIANT MEASURES FOR CONTINUOUS TRASFORMATIONS
3.1 MIXING
Definition:
Let T be a measure-preserving transformation of the probability space (X, โฌ, m) .
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(i)T is weak- mixing if โ๐ด, ๐ต โ โฌ
lim๐โโ
1 n m TโiA โฉ B โ m A m(B)
nโ1
i=0
= 0.
(ii) T is strong- mixing if โ๐ด, ๐ต โ โฌ
lim๐โโ
m TโnA โฉ B = m A m(B).
Remarks:
1. Every strong- mixing transformation is weak- mixing and every weak- mixing
transformation is ergodic . This is because if an is a sequence of real numbers
thenlim๐โโ an = 0 implies
lim๐โโ
1 n ai
nโ1
i=0
= 0
And this second condition implies
lim๐โโ 1
n ainโ1i=0 = 0.
2. Intutive descriptions of ergodicity and strong-mixing can be given as follows.To say T is
strong- mixing means that for any set ๐ด the sequence of sets TโnA becomes,
asymptotically, independent of any other set ๐ต. Ergodicity means TโnA becomes independent
of ๐ต on the average, for each pair of sets ๐ โ
Examples:
1. An example of an ergodic transformation which is not weak-mixing is given by a rotation
T z = az on the unit circle ๐พ.If ๐ด and ๐ต are two small intervals on ๐พ then TโiA will be
disjoint from ๐ต for atleast half of the values of i so that
1 n m TโiA โฉ B โ m A m(B)
nโ1
i=0
โฅ1
2m A m(B)
for large n . From this one sees that , intuitively, a weak-mixing transformation has to do some
โstretchingโ.
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2. There are examples of weak-mixing T which are not strong-mixing. Kakutani has an example
constructed by combinatorial methods and Maruyama constructed an example using Gaussian
processes. Chacon and Katok and Stepin also have examples.If(X, โฌ, m) is a probability space, let
๐(๐) denote the collection of all invertible measure-preserving transformations of (X, โฌ, m). If
we topologize ๐(๐) with the โweakโ topology ,the class of weak- mixing transformations is of
second category while class of strong- mixing transformations is of first category .So from the
view point of this topology most transformations are weak-mixing but not strong-mixing.
The following thoremgives a way of checking the mixing properties for examples by reducing
the computations to a class of sets we can manipulate with. For example, it implies we need only
consider measurable rectangles when dealing with the mixing properties of shifts.
Theorem 3.1.1:
Let (X, โฌ, m)be a measure space and let โ be a semi-algebra that generates โฌ. Let ๐: ๐ โ ๐ be a
measure-preserving transformation. Then
(i)๐ is ergodic iff โ๐ด, ๐ต โ โ
lim๐โโ
1 n m TโiA โฉ B
nโ1
i=0
= m A m B ,
(ii)T is weak- mixing iff โ๐ด, ๐ต โ โ
lim๐โโ 1
n m TโiA โฉ B โ m A m(B) nโ1i=0 = 0,and
(iii)T is strong- mixing if โ๐ด, ๐ต โ โฌ
lim๐โโ
m TโnA โฉ B = m A m(B).
Proof:
Since each member of the algebra,๐(โ), generated by โcan be written as a finite disjoint union
of members of โ it follows that if any of the three convergence properties hold for all members
of โ then they hold for all members of ๐(โ).
Let ๐ > 0 be given and let ๐ด, ๐ต โ โฌ. Choose ๐ด๐ , ๐ต๐ โ ๐(โ) with
๐(๐ด โ ๐ด๐) < ๐and๐(๐ต โ ๐ต๐) < ๐ .For any ๐ โฅ 0,
TโiA โฉ B โ Tโi๐ด๐ โฉ ๐ต๐ โ TโiA โฉ Tโi๐ด๐ โช ๐ต โ ๐ต๐ ,
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so we have ๐ TโnA โฉ B โ Tโi๐ด๐ โฉ ๐ต๐ < 2๐, and therefore
๐ TโiA โฉ B โ ๐ Tโi๐ด๐ โฉ ๐ต๐ < 2๐.
Therefore
๐ TโiA โฉ B โ m A m B โค ๐ TโiA โฉ B โ ๐ Tโi๐ด๐ โฉ ๐ต๐
+ ๐ Tโi๐ด๐ โฉ ๐ต๐ โ m ๐ด๐ m ๐ต๐
+ m ๐ด๐ m ๐ต๐ โ m A m ๐ต๐
+ m ๐ด๐ m ๐ต๐ โ m A m B
< 4๐ + ๐ Tโi๐ด๐ โฉ ๐ต๐ โ m ๐ด๐ m ๐ต๐ .
This inequality together with the known behaviour of the right hand term proves (ii) and (iii). To
prove (i) one can easily obtain
1 n ๐ TโiA โฉ B โ m A m B
nโ1
i=0
< 4๐ + 1 n ๐ Tโi๐ด๐ โฉ ๐ต๐ โ m ๐ด๐ m ๐ต๐
nโ1
i=0
and then use the known behaviour of the right hand side.
As an application of this result we shall prove the result about ergodicity of Markov shifts .To do
this we shall use the following:
Lemma 3.1.1:
Let ๐ be a stochastic matrix, having a strictly positive probability vector ๐ with ๐๐ = ๐. Then
๐ = lim๐โโ(1
๐) ๐๐๐โ1๐=0 exists. The matrix ๐ is also stochastic and ๐๐ = ๐๐ = ๐. Any
eigenvector of ๐ for the eigen value 1 is also an eigen vector of ๐. Also๐2 = ๐.
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Theorem 3.1.2:
Let T denote the ๐, ๐ Markov shift( either one-sided or two-sided).We can assume ๐๐ > 0 for
each ๐ where ๐ = ๐0, ๐1, โฆ , ๐๐โ1 . Let ๐ be the matrix obtained in Lemma. The following are
equivalent:
(i) T is ergodic.
(ii) All rows of the matrix ๐ are identical.
(iii) Every entry in ๐ is strictly positive.
(iv) ๐is irreducible.
(v) 1is a simple eigen value of ๐.
3.2 INVARIANT MEASURES FOR CONTINUOUS TRASFORMATIONS
The space ๐ denotes a compact metrisable space and ๐ will denote a metric on ๐ . The
๐-algebra of Borel set by โฌ ๐ . So โฌ ๐ is the smallest ๐-algebra containing all open subsets
of ๐ and smallest ๐-algebra containing all closed subsets of ๐ . ๐ ๐ denote the collection of all
probability measures defined on the measurable space ๐, โฌ ๐ and the members of ๐ ๐
are called Borel probability measures on ๐.Each ๐ฅ โ ๐ determines a member ๐ฟ๐ฅ of ๐ ๐
defined by
๐ฟ๐ฅ ๐ด = 1 ๐๐ ๐ฅ โ ๐ด0 ๐๐ ๐ฅ โ ๐ด.
๐ ๐ is a convex set where ๐๐ + 1 โ ๐ ๐ ๐๐ ๐๐๐๐๐๐๐ ๐๐ฆ
๐๐ + 1 โ ๐ ๐ ๐ต = ๐๐ ๐ต + 1 โ ๐ ๐ ๐ต if ๐ โ 0,1 .
Theorem 3.2.1
A Borel probability measure ๐ on a metric space ๐ is regular.
Corollary 3.2.1:
If ๐: ๐ โ ๐ is a continuous transformation of a compact metric space ๐ then ๐ ๐, ๐ is non-
empty.
The following are some properties of ๐ ๐, ๐ .
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Theorem 3.2.2:
If ๐ is a continuous transformation of the compact metric space ๐ then
(i) ๐ ๐, ๐ is a compact subset of ๐ ๐ .
(ii) ๐ ๐, ๐ is convex.
(iii) ๐is an extreme point of ๐ ๐, ๐ iff ๐ is an ergodic measure-preserving transformation of
๐, โฌ ๐ , ๐ .
(iv) If ๐, ๐ โ ๐ ๐, ๐ are both ergodic and ๐ โ ๐ then they are mutually singular.
Remark 3.2.2:
Since ๐ ๐, ๐ is a compact convex set each member of ๐ ๐, ๐ can be expressed in terms of
ergodic members of ๐ ๐, ๐ . If ๐ธ ๐, ๐ denote the set of extreme points of ๐ ๐, ๐ then for
each ๐ โ ๐ ๐, ๐ there is a unique measure ๐ on the borel subsets of the compact metrisable
space ๐ ๐, ๐ such that ๐ ๐ธ ๐, ๐ = 1 and โ๐ โ ๐ถ ๐
โซ๐๐ ๐ฅ ๐๐ ๐ฅ = โซ
๐ธ ๐ ,๐ โซ
๐๐ ๐ฅ ๐๐ ๐ฅ ๐๐(๐).We write ๐ = โซ
๐ธ ๐ ,๐ ๐๐๐(๐) and call this the
ergodic decomposition of ๐. Hence every ๐ โ ๐ ๐, ๐ is a generalized convex combination of
ergodic measures.
3.3 INTERPRETATION OF ERGODICITY AND MIXING
Let ๐: ๐ โ ๐ be a continuous transformation of a compact metric space. We say
๐ ๐ ๐(๐, ๐) is ergodic or weak-mixing or strong-mixing if the measure-preserving transformation
has corresponding property. ๐is ergodic iff โ๐, ๐ โ ๐ฟ2(๐)
1
๐ f Ti x g x
๐โ1
๐=0
๐๐ ๐ฅ โ ๐๐๐ ๐ ๐๐.
Lemma 3.3.1:
Let ๐ ๐ ๐(๐, ๐).Then
(i)๐ is ergodic iff โ๐ โ ๐ถ ๐ โ๐ โ ๐ฟ1(๐)
1
๐ โซ f Ti x g x ๐โ1๐=0 ๐๐ ๐ฅ โ โซ๐๐๐ โซ๐ ๐๐.
(ii)๐ is strong-mixing iff โ๐ โ ๐ถ ๐ โ๐ โ ๐ฟ1(๐)
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๐(Ti x )g x ๐๐ ๐ฅ โ ๐๐๐ ๐ ๐๐.
(iii)๐ is weak-mixing iff there is a set ๐ฝ of natural numbers of density zero such that
โ๐ โ ๐ถ ๐ โ๐ โ ๐ฟ1 ๐ ,
lim๐ฝโ๐โโ โซ๐(Ti x )g x ๐๐ ๐ฅ โ โซ๐๐๐ โซ๐ ๐๐ .
Proof:
(i)Suppose the convergence condition holds and let ๐น, ๐บ โ ๐ฟ2(๐). Then ๐บ โ ๐ฟ2(๐) so, โ๐ โ ๐ถ ๐
1
๐ โซ f Ti x g x ๐โ1๐=0 ๐๐ ๐ฅ โ โซ๐๐๐ โซ๐ ๐๐ .
Now approximate ๐น in ๐ฟ2(๐) by continuous functions to get
1
๐ โซ๐น Ti x x ๐โ1๐=0 ๐บ๐๐ ๐ฅ โ โซ๐น๐๐ โซ๐บ๐๐ .
Now suppose ๐ is ergodic. Let ๐ โ ๐ถ ๐ . Then ๐ โ ๐ฟ2 ๐ so if โ ๐ฟ2 ๐ we have
1
๐ โซ f Ti x h x ๐โ1๐=0 ๐๐ ๐ฅ โ โซ๐๐๐ โซ ๐๐ .
If ๐ โ ๐ฟ1(๐) then by approximating ๐ in๐ฟ1(๐) by โ ๐ฟ2 ๐ we obtain
1
๐ โซ f Ti x g x ๐โ1๐=0 ๐๐ ๐ฅ โ โซ๐๐๐ โซ๐ ๐๐ .
Similarly we can prove part (ii) and (iii).
Theorem 3.3.1:
Let ๐ be a continuous transformation of a compact metric space. Let ๐ ๐ ๐(๐, ๐).
(i)๐ is ergodic iffwhenever ๐๐๐(๐)and ๐ โช ๐ then
1
๐ ๐ ๐๐ โ ๐๐โ1๐=0 .
(ii)๐is strong-mixing iffwhenever ๐๐๐(๐)and ๐ โช ๐ then๐ ๐๐ โ ๐ .
(iii)๐ is weak-mixing iff there is a set ๐ฝ of natural numbers of density zero such thatwhenever
๐๐๐(๐)and ๐ is ergodic then
lim๐ฝโ๐โโ
๐ ๐๐ โ ๐ .
Proof:
(i)We use the ergodicitycondition of the above lemma .
Let ๐ be ergodic and suppose ๐ โช ๐ , ๐๐๐(๐) .Let ๐ = ๐๐ ๐๐ ๐ ๐ฟ1(๐) . If ๐ โ ๐ถ ๐ then
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๐ ๐ 1
๐ ๐ ๐๐
๐โ1
๐=0
=1
๐ ๐ โ
๐โ1
๐=0
Ti dm
=1
๐ f Ti x g x
๐โ1
๐=0
๐๐ ๐ฅ
โ ๐๐๐ ๐ ๐๐
= ๐๐๐ 1๐๐
= ๐ ๐๐ .
Therefore,1
๐ ๐ ๐๐ โ ๐๐โ1๐=0
We now show the converse.
Suppose the convergence condition holds. Let ๐ โ ๐ฟ1(๐) and ๐ โฅ 0 . Define ๐๐๐(๐) by
๐ ๐ต = ๐ โซ๐ ๐๐
where ๐ = 1(โซ๐ ๐๐)
.
Then if ๐ โ ๐ถ ๐ we have ,by reversing the above reasoning,
1
๐ โซ f Ti x g x ๐โ1๐=0 ๐๐ ๐ฅ โ โซ๐๐๐ โซ๐ ๐๐ .
If ๐ โ ๐ฟ1(๐) is real-valued then ๐ = ๐+ โ ๐โ where ๐+, ๐โ โฅ 0 and we apply the above to ๐+
and ๐โ to get the desired condition of the lemma for ๐ .The case of the complex-valued ๐
follows .Similarly, we can prove part (ii) and (iii).
3.4 UNIQUE ERGODICITY
Definition3.4.1:
A continuous transformation ๐: ๐ โ ๐ is a compact metrisable space ๐ is uniquely ergodic if there is only one ๐ invariant Borel probability measure on ๐,i.e, ๐(๐, ๐) consists of one point.
If ๐is uniquely ergodic and ๐ ๐, ๐ = {๐} then ๐ is ergodic because it is an extreme point of
๐ ๐, ๐ .
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Unique ergodicity is connected to minimality by:
Theorem3.4.1:
Suppose ๐: ๐ โ ๐ is a homeomorphism of a compact metrisable space ๐ .Suppose ๐ is
uniquely ergodic and ๐ ๐, ๐ = {๐} . Then ๐ is minimal iff ๐(๐) > 0 for all non-empty open
sets ๐.
Proof:
Suppose ๐ is minimal. If ๐ is open , ๐ is open, ๐ โ โ , then ๐ =โช๐=โโโ ๐๐ ๐ , so if ๐ ๐ = 0
then ๐ ๐ = 0, a contradiction.
Conversely, suppose ๐(๐) > 0for all open non-empty ๐. Suppose also that ๐ is not
minimal.There exists a closed set ๐พ such that ๐๐พ = ๐พ, โ โ ๐พ โ ๐. The homeomorphism ๐ ๐พ has
an invariant Borel probability measure ๐๐พ on ๐พ. Define ๐ on ๐ by ๐ ๐ต = ๐๐พ(๐พ โฉ ๐ต)for all Borel
sets ๐ต. Then ๐ โ ๐ ๐, ๐ and๐ โ ๐ because ๐(๐ ๐พ ) > 0 , as ๐
๐พ is non-empty and open,
while ๐ ๐ ๐พ = 0. This contradicts the unique ergodicity of ๐.
Example 3.4.1:
The map ๐: ๐พ โ ๐พ given by
๐ ๐2๐๐๐ = ๐2๐๐๐2, ๐ โ 0,1 ,
is an example of a uniquely ergodic homeomorphism which is not minimal. The point 1 โ ๐พ is a
fixed point for ๐ and ฮฉ ๐ = 1 so that ๐ ๐พ, ๐ = {๐ฟ1}.
Remark 3.4.1:
Results about unique ergodicity known before are given in Oxtoby. More recent results of Jewett
and Kringer imply that any ergodic invertible measure-preserving transformation of a Lebesgue
space is isomorphic to a minimal uniquely ergodic homeomorphism of a zero dimensional
compact metrisable space. In particular there are minimal uniquely ergodic homeomorphisms
with any prescribed non-negative real number for their entropy. Hahn and Katznelson had found
minimal uniquely ergodic transformations with arbitrarily large measure-theoretic entropy.
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Theorem 3.4.2:
Let ๐: ๐พ โ ๐พ be a homeomorphism with no periodic points. Then ๐is uniquely ergodic.
Moreover
(a) there is a continuous surjection ๐: ๐พ โ ๐พ and a minimal rotation ๐: ๐พ โ ๐พ with
๐๐ = ๐๐. The map ๐ has the property that for each ๐ง โ ๐พ, ๐โ1(๐ง) is either a point or a closed
sub-interval of ๐พ.
(b) If ๐ is minimal the map ๐ is a homeomorphism so that every minimal homeomorphism of
๐พ is topologically conjugate to a rotation.
Example 3.4.2:
H . Furstenberg constructed an example of a minimal homeomorphism ๐: ๐พ2 โ ๐พ2 of the two
dimensional torus which is not uniquely ergodic. The example preserves Haar measure and has
the form ๐ ๐ง, ๐ค = (๐๐ง, ๐(๐ง)๐ค) where {๐๐}โโโ is dense in ๐พ and ๐: ๐พ โ ๐พ is a well chosen
continuous map.
4. APPLICATIONS
NORMAL AND SIMPLY NORMAL NUMBERS
An important application of Birkhoff Ergodic Theorem is the study of normal numbers. A number
is said to be simply normal to the base ๐ if,for ๐ โ 0,1,2,3, โฆ , ๐ โ 1 , the average frequency of
occurrence of ๐ in the base ๐ expansion of ๐ฅ is 1 ๐ . In otherwords, a number is normal to the
base ๐ if,for every sequence of ๐ digits, ๐ โ ๐,the average frequency of occurrence of that
sequence in the ๐-expansion of ๐ฅ is 1 ๐๐ .
Every number which is normal to base ๐ is simply normal to the base ๐ by its definition.This
result and definition is due to Borel. Borel further proved in 1909 that except for a subset of
measure zero, every ๐ฅ โ 0,1 is normal. We will prove this result in this section using ergodic
theory.
We formalize our definition for base ๐ series expansion of ๐ฅ โ 0,1 by defining a
transformation
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๐ ๐ฅ = ๐๐ฅ ๐๐๐ 1 =
๐๐ฅ ๐ฅ โ 0,
1
๐
๐๐ฅ โ 1 ๐ฅ โ 1
๐,2
๐
โฎ
๐๐ฅ โ ๐ โ 1 ๐ฅ โ ๐ โ 1
๐, 1
Our expansion of base ๐ results from iterating this map ๐ . We define
๐1 ๐ฅ = ๐๐ฅ and
๐๐ ๐ฅ = ๐๐๐โ1๐ฅ = ๐1 ๐
๐โ1๐ฅ .
From our definition, we have
๐ฅ =๐1๐
+๐ ๐ฅ
๐
And we find that
๐ฅ =๐1๐
+๐ ๐ฅ
๐
=๐1๐
+๐2๐2
+๐2
๐2
=๐1๐
+๐2๐2
+ โฏ +๐๐๐๐
+ โฏ
= ๐๐ ๐ฅ
๐๐
โ
๐ =1
and this series converges to ๐ฅ.
Definition 4.1
A number ๐ฅ โ 0,1 is simply normal to base ๐ if for every ๐ โ 1,2, โฆ . , ๐ โ 1 ,
lim๐โโ๐ ๐ ,๐
๐=
1
๐.
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Definition 4.2
A number ๐ฅ โ 0,1 is normal to base ๐ if for every ๐-length sequence of digits ๐1๐2 โฆ๐๐ ,
where ๐1, ๐2 , โฆ , ๐๐ โ 1,2, โฆ . , ๐ โ 1 ,
lim๐โโ1
๐# ๐ 1 โค ๐ โค ๐ ๐๐๐ ๐๐ ๐ฅ = ๐1, โฆ , ๐๐+๐โ1 ๐ฅ = ๐๐ =
1
๐๐.
We illustrate as follows.
Theorems 4.1(Borelโs Theorem on Normal Numbers):
Almost all numbers in 0,1 are normal to the base 2, i.e., for a.e. ๐ฅ โ 0,1 the frequency of 1โs in
the binary expansion of ๐ฅ is 1 2 .
Proof:
Let ๐: 0,1 โ 0,1 be defined by ๐ ๐ฅ = 2๐ฅ ๐๐๐1. We know that ๐ preserves Lebesgue
measure ๐ and is ergodic. Let ๐ denote the set of points of 0,1 that have a unique binary
expansion. Then ๐ has a countable complement so ๐ ๐ = 1.
Suppose ๐ฅ =๐1
2 +๐2
22 + โฏ has a unique binary expansion. Then
๐ ๐ฅ = ๐ ๐1
2 +๐2
22 + โฏ =๐2
2 +๐3
22 + โฏ
Let ๐ ๐ฅ = ๐ณ
1
2,1
๐ฅ . Then
๐ ๐๐๐ฅ = ๐ ๐๐+1
2 +๐๐+2
22 + โฏ = 1 ๐๐๐ ๐๐+1 = 10 ๐๐๐ ๐๐+1 = 0.
Hence if ๐ฅ โ ๐ the number of 1โs in the first n digits of the dyadic expansion of ๐ฅ is
๐ ๐๐๐ฅ .๐โ1๐=0 Dividing both sides of the equality by n and applying the ergodic theorem we get
1 ๐ ๐ ๐๐๐ฅ โ ๐ณ
1
2,1
๐๐ = 1 2
๐โ1
๐=0
๐. ๐.
This says that the frequency of 1,s in the binary expansion of almost all primes is1 2 .
CONCLUSION
This paper aimed to bring out the importance of ergodic theory in mathematics. Ergodic
theory is one of the few branches of mathematics which has changed radically during the last two
decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily
with averaging problems and general qualitative questions, while now it is a powerful amalgam of
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methods used for the analysis of statistical properties of dynamical systems. For this reason, the
problems of ergodic theory now interest not only the mathematician, but also the research
worker in physics, biology, chemistry e.t.c.
The main principle which adhered to from the beginning was to develop the
approaches and methods of ergodic theory in the study of numerous concrete examples. Because
of this, the paper contains the description of the fundamental notions of ergodicity and mixing.
In recent years there have been some fascinating interactions of ergodic theory with
differentiable dynamics, Von Neumann algebras, probability theory, statistical mechanics, and
other topics. This paper tries to tackle many difficulties regarding ergodic theory and its
applications.
REFERENCES
1. Walters , Peter, AN INTRODUCTION TO ERGODIC THEORY, Springler, 1982.
2. Coudene, Yves, ERGODIC THEORY AND DYNAMICAL SYSTEMS, Springler, 2013.
3. Nadkarni, M.G., BASIC ERGODIC THEORY, Springler,2013.
4. Cornfeld, I.P. ,Fomin, S.V., Sinai, Ya.G., ERGODIC THEORY, New York: Springler-Verlag, 1982.
5. Billingsley, P., ERGODIC THEORY AND INFORMATION, Wiley, New York, 1965.
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