basic ergodic theory notes

Eduard Emel'yanov

INVITATION TO ERGODIC THEORY

PrefaceThe ergodic theory goes back to the end of 19th centuryand has roots in physics (in particular in astronomy andin gas mechanics). Nowdays it is one of basic techniques inprobability theory, measurable dynamics, and in many otherareas of mathematics.

The main aim of this course is to give an introduction tobasic concepts of ergodic theory such as recurrence, ergod-icity, mixing, weak mixing, etc. It requires some knowledgeof measure theory and Lebesgue integration. We include ashort presentation of these topics in Chapter 1. Althoughthe material of this chapter is standard and may be found inmany textbooks (see, for example, 17,4,9,10]) we includefew proofs of interesting elementary theorems there. Weconsider recurrence and ergodic transformations of measurespaces, and study selected classical examples i n Chapter 2.

Then, in Chapter 3, we prove the Birkhoff individual ergodictheorem and the von Neumann mean ergodic theorem. Inthe last chapter, Chapter 4,, we discuss Frobenius - Perronoperators, mixing, exactness, and weak mixing.

We emphasize the introductionary character of this one-semester graduate course and send the reader for delicate,difficult and interesting topics of ergodic theory to textbooks

[3], l5l , l7l, [11]

2

Contents

Elements of Measure Theory

1.1. o-Algebras1.2. Measures1.3. Lebesguelntegration

Elements of Measurable Dynamics

2.1. Examples of tansformations2.2. Recurrence2.3. Ergodic Tbansformations

Ergodic Theorems

3.1. The Birkhoff Ergodic Theorem3.2. The von Neumann Mean Ergodic Theorem3.3. Applications of Ergodic Theorems

Related Topics

4.1. Flobenius - Perron operators4.2. Mixing

Bibliography

Chapter 1

Elements of Measure Theory

In this chapter, we outline briefly several essential conceptsof the measure theory and the Lebesgue integration. Forexhaustive presentation of these topics, we refer to standardtextbooks [1, 4, 9, 10].

1.1 o-Algebras

The theory of o-algebras plays a significant role in the prob-ability theory and in the ergodic theory.

1.1.1 Algebras and o-algebras of sets

Definition 1.1.1 A collect'ion\ of subsets of a set X iscalled an algebra i,fa)X€I;b)A€I+X\AeI;c)A,,BeI+AUBeI.

CHAPTER 1. ELEMENTS OF MEASURE THEORY

It is an easy exercise to show that any algebra is closed underoperations -4 n B and,4A B - (A \ B) U (B \ A)

Definition 1.L.2 An algebra D 'is called a o-algebra,f UAn €D yi,elds for any countable fami,ly {An}n g I.

k

As in the previous exercise, it is easy to show that any o-algebra is closed under countable intersections. A bit moredifficult (but stil trivial) exercise is to show that an algebra,which is ciosed under countable unions of pairwise dis.jointsequences of sets, is a o-algebra.

A simple example of a o-algebra is the collection P (X) ofali subsets of a nonempty set X.As an example of algebra which is not a o-algebra, one mayconsider the collection A(T,R) of all intervals (o,,b), (o,bl,lo,b),, lo,b1, (-oo, b) , (o,+oo) in IR and their finite unions.

Exercise 1.1. 1 Show that the fami,Iy of all countableand all co-countable subsets of a non-empty set,is a o-algebra. What would be 'in the case i,f finite subsets ,in-

stead of countable subsets are considered.

Given a non-empty family F of subsets of a non-empty set X,the algebra A(F) generated by F is the intersection ofall algebras containing F (at least one such an algebraexists, the algebra P (X)). The definition of the o-algebraA"(F) generated by F is similar.

1.1. r-ALGEBRAS 5

Exercise 1 .L.2 Show that A(7, IR) - A(F), whereF - {(-oo, bl}a.o u {{b}}a.o.

The o-algebra A,(A(T,R)) is called the Borel o-algebraof subsets of lR. More generally, the Borel o-algebra ofa topological space (X, r) is the o-algebr a Ao(r) generatedby the collection r of all open subsets of X .

It is often a challenging problem to describe the o-algebraA"(F) generated by a family .F in a more constructive waythan just the intersection of all a-algebras containing F.

In the next two subsections we study several important re-sults about o-algebras and recommend the reader to com-plete all details of the proofs below, or to consult textbooksIike [1 , 4, 6, I7].

1.L.2 Dynkin systems (optional)

Definition 1.1.3 A collect'ion K of subsets of X ,is sa,idto be o zr-system ,f i,t 'is closed underintersect,ion of angtwo of its sets, and A e K.

Denote by r(F) the zr'-system generated by F gP6)

6 CHAPTER 1. ELEMENTS OF MEASURE THEORY

Definition 1.L.4 A collect'ion D g P(X) is called a,

Dynkin system i/1) X eD;2)AeD+X\AeD;3) {A*}t'.xgD k An)Ap:aforklp impl'ies

UAn e D.k

Denote by D (F) the Dynkin system generated byF EP6)

Theorem 1 If K be a n-system then A"(K) -D(K).

Proof: It is obvious that D(K) q A"@) is a Dynkinsystem. We show that A,(K) g D(K)Note that any Dynkin system, which is a a"-system, isa o-algebra. So, it is enough to show that D(K) isa n'-system.

Let A € D(K) and consider the following family:

DA: {B e D(K): AnB e D(K)}.

The collection D,q, is a Dynkin system. Indeed, the condition1) of the last definition is obvious. Let {Bp}p q Da andBn) B^ : o when k I *. Then {An B*)n eD(K) is apairwise disjoint family. Hence

Ute n ail - An U Bn e D(K)

1.1. r-ALGEBRAS

and thenBn e Dn.

Then the condition (3) is satisfied.

To check the conditio" (2), take B e Da. By

An(x\ B):x\((An B)u (x\ A)),

Exercise 1.1.3 Checle the last formula carefully.

We obtainAn(x\B)eDa

since A n B € D(K), X \ A € D(K), and since(AnB) u(x\ A):sThus Dars a Dynkin system for any A e D(K)

Exercise 1.L.4 ShowthatAe K + K gDa.(ntnt Be K+Bn AeKcD(K)+aenn)

Let A € K. Using K g Dt, we obtainD(K) g Da

-c D(K), and hence D(K) - Da for every

A e K. Consider the foilowing implications:

AeK+DA:D(K)+li,f A e K and B e D(K) - DA, then A)B e D(K)I =+

lx q Dn e D@) i,f B e D(K)) +DB : D(K) (VB € D(K)) =+

A)B eD(K) (VA,B e D(K)).The last condition means that D(K) is closed under theintersection, what is required. I

Uk


1.1.3 The z--.\-Theorem and the Monotone Class Theorem(optional )

Let us consider now two more results of the same naturewhich are known as the zr-)-Theorem and the MonotoneClass Theorem.

Definition 1.L.5 A collect'ion L C P(X) 'is calleda ,\-system or \-class fo)XeL;b)A,Be L, AgB+B\AeL;c){An}ngL, AnIA+AeL.Denote by )(F) the ,\-system generated by F gP(X)

Theorern 2 (zr-,\-Theorem)U F is a r-system then A"(F) -

^(.F).Proof: Obviously .\(.F) g A"(F)To prove )(f) ) Ao(f), it suffices to show that )(f) is ao-algebra. It is enough to show that )(.F) is closed undercountable intersections. (WhA? Show thislConsider

)'(.r) -{Ae )(r) :AnBe^(.F) VBe f).Then

^tV) is a )-system (Why? Show this!) containing

F. Hence )t("r) - )(.r).Consider now

^rV)-{A€.\(r) :AnBe^(F) VBe )(-r)}

1.1. r-ALGEBRAS

Then ^r(F)

is (again as above) a ,\-system containi ng F(Hint: start w'ith x(F) instead of f and remark that)(^(4) - ^(r)).

Hence F e ^r(f)

g ,\(4 and there-fore )2 (F) - ^(F).

The last equality means that ,\(.F) isclosed under countable intersections, what is required). n

Definition 1.1.6 A fami,lA M g P(X) i,s called a mono-tone class i/i,)x#a;i,i,) {An}r g M, An I A + A e M;iii) {Br}r I M, Bn I B + B e M.

Denote by M(F) the monotone class generated byF CP6)

Exercise 1.1.5 Show that if a monotone class M ,is analgebra tlr,en M is a o-algebra.

Theorem 3 (Monotone Class Theorem)If A 'is an algebra then A"(A) : M(A).

Proof: Obviously M(A) S A"(A). To show the converseA"(A) g M(A), it is enough to prove that M(A) is analgebra.

For each G g X, define

MG:{BgX: B\G, BuG, G\B€M(/)}It is obvious that Ms is a monotone class (just becauseM(A)is)


Exercise 1.1.6 Show that E e A+ Ag Ms.

Thus M(A) e Mn for all E e A. Therefore

C e Mn (VE € A,C e M(A)),

which is equivalent to E e Mc (WhA? Show thi,sl.

So, for any C e M(A)), M(/)) q Ms (since A g Msand Ms is a monotone class). The condition

M(A) e Mc e M(A) (vc € M(/))means that for every B, C e M(A):

B\C, BuC, C\B €M(A).Together with the fact that X € M(A)., this implies thatM(A) is an algebra. The proof is complete. n

1.2. MEASURES

L.2 Measures

Definition 1.2.L Let A be a o-algebra of subsets of X.A funct'ion pr,: A- nR - R U{+*} ,is called, o measure,fo) p(a) - 0;

b) p(A))0forallA€A;c) t-r (? r-) : ? t @n) for any countabte fam,ity {An}nof pairwise disjo'int sets belong'ing to A.

In this course) we are interested mostly in "small measures" :

probabilistic (p(X) - 1), finite (p(X)_. oo),

or o-finite (={&}8, q "4 such that X - [J X" and

lr(X,) ( oo for each n). n':1

Definition 1.2.2 A tri,ple (X,A, p,) it called a measurespace.

Except of counting measures on countable sets, construct-ing a measure on a o-algebra could be a difficult task. Weshall not go deeply in details and only mention key steps ofconstruction of the Lebesgue measure on IR.

L.z.L Lebesgue measure on IR

First of all, we need a large enough a-algebra (containingthe Borel algebra 6(R)) otr which our measure will live.

11

12 CHAPTER 1. ELEMENI:S OF MEASURE THEORY

Secondly, this measure should agree with a certain functionon a certain subset of 6(R) (for example, with the length onthe collection J g 6(R) of all intervals).

For this purpose, w€ extend the length I . J -+ R+ -IR+ U {+*} to a function \ . P(R) * R1, and then takethe restriction of ) on a certain o-algebra D ) B(R), otrwhich ,\ acts as a measure. Some of details follows.

)(A) - inr {: ren): 11, e r k Asp.r}

I - {A, )(E) : \(E.A)+)(tr.A") for all E e 2(R)},where we denote A" - R \ A.

The classical Caratheodory theorem says that t is a o-algebra containing 6(R) and )(1) - l(I) for every I € J.Moreover, D is complete w.r.t. ) (i.e. )(B) - 0 =+ B e I)and ,\lr is a measure.

Remark that the Caratheodory theorem holds true if we startwith any countabiy additive nonnegative function on an al-gebra of sets (not necessary with the length of intervals).

Definition 1.2.3 The o-algebra D constructed abouecalled a Lebesgue algebra, and the n'Leasure )1"called o Lebesgue measure.

It is worth to note that the Caratheodory theorem is ratherdeep result and that the ideas that had led to the Lebesguemeasure space (R, I(lR),

^) go back to the end of 19th cen-

tury.

,is

,is

1.2. MEASURES

Remark that the Lebesgue algebra D(R) is strictly biggerthan the Borel algebra 6(lR). It can be shown (for example,by using of Monotone Class Theorem) that the cardinality ofthe Borel algebra Card(6(R)) is c - 2N. On the other hand,the ternary Cantor set C is Borel measurable, Card(C) : c,

and )(C) : 0. Therefore by the completeness of D w.r.t. ),all subsets of C belong to I and thus

Card(D) ) Cu'd(P(C)) :2') c - Card(B(R)).

Although Card(I) - Card(P(R)), the o-algebra I is aproper subalgebra of 2(R). By using the Axiom of Choice,it is rather easy to construct an example of a subset of IR.

which is not Lebesgue measurable.

L.2.2 Isomorphisms of measure spaces (optional)

Two measure spaces (X, A,,p) and (X' , A , p') are said to beisomorphic mod 0 if there exist measurable sets Xo e X ,

XI S X' of full measure (i.e., p,(X \Xr) - p'(X'\X6) -0) and a bijection Xo I ,'rsatisfying1) A € A' 1"6, <+ 0-'@) € Alxo,z) p(d-'(A)) - t''(A)YA e AlroWe call such a bijection O an isomorphism mod 0 or justisomorphism.

Example 1 .2.L Let F e [0, 1] be a closed set such that)(F) > 0. Defi,ne a n'Leasure p on Dlp by

p(A) - I(,4) l^@) (A e tlr).

13


Defineo,rnapQ:F---+ [0, 1] by

_ )(F n [0, r))XF)

d@)

It i,s not ltard to uerify that (f, f le,lt) 'is 'isomorpltic to(10, 1], Illo,tl, )) bg the 'isomorphi,sm Q.

More interesting fact is that any nonatomic completeseparable measure space is isomorphic to (1, Ilr, )), where1 g R is an interval (bounded or unbounded).

L.2.3 Lebesgue measure spaces (LMS)

Definition 1.2.4 A measure space (X,A, p) 'is sa'id tobe a canonical Lebesgue measure space, or s'implyLMS i/1) (X, A, p) 'is o -fini,te;2) X - D I I'is a d'isjoi,nt un'ion of a countable set Dand an'interual / q R such that (L,Alr, p) : (I ,t, )).The component (D, Alo,p) is called a canonical atomicLMS. The component (I ,Alr,F) (which is nothing else

than the interval -I equipped with the Lebesgue measure .\)is called a canonical nonatomic tMS.

Definition 1 .2.5 A complete measure space (X, A, p) iscalled a Lebesgue space if it 'is 'isomorph'ic mod 0 to acanon'ical LMS.

Remark that in the ergodic theory, Lebesgue spaces are ofmain interest among all measure spaces.

1.2. MEASURES

I.2.4 Approximation with semirings (optionar)

Definition 1.2.6 A nonernpty r-system R g P(X) iscalled a semiring on X if , for any A, B e R, the;e erist

disjoi,nt sets Et,...,En €R such that A \ B - lJ Er.k:l

As an example of semiring, one may consider the collectionJ(R) of all intervals in IR.

The following proposition is almost obvious.

Proposition L.z.L Let R be a sem'iring. If A - U An,n:I

where An e R, then A - i) ,r, where Cn e R.k:L

Proof: Define a sequence {8"}" by 81 - At, and for

n ) 1, Bn- An\(Aru. .. uA,-t). Then A - 1l f,. The

only thing that is left to write each B, asairi#":i union ofsets belonging to R. We show this by induction. Obviously,

m,p

81 € R. Assume that B, - l) D'u, where Den € R, for

p: I,,2,...1r1. Our aim is to Jfrl* that Bn+rsatisfies thesame property.

nnBy the construction of {Bn}n, U Ai : I) Bi. Hence

j:7 j:I

Bn+r:An+r\U Aj:An+r \U Bj:

15

j:I j:1


nnpj

-nAn+t\Bi-nl)o'-j--1' j:l k:1

for som "

Etk e R. We may also suppose that all p1 - / by

adding, if it is necessary, some E* - a. Thus

nnl'bn+t-nLlr'r-i:L k:7

lJ t E/,rn tr'*rn... n 4,, kt,kz,...,k, € {7,2,..., /}}which is nothing else than a disjoint union of elements of R.

By the induction, all Bn have such representation, whichcompletes the proof. n

Let us mention the following useful approximation propertywhich follows directly from Definition I.2.7 and Proposi-tion 7.2.7.

Proposition L.2.2 (First Littlewood Principle) Let(X, A, p) be a nxeasure space wi,th a suffic'ient sem'iring

Definition 1.2.7 Let (X,A, p) be a measure space. Asem'iring C g A of sets o/ finite measure is sa,id to be

a suffi.cient semiring for (X,A,p,) xf for euery A e A:

,r {i te): Ae|", , ci. c}t j:l i:7 )

1.2, MEASURES

C. Let A e A, p(A)

erists a fin'ite disjoi,nt un'ion

such tlt at

17

Cp of sets Cn € C

and let ep

Ao: lJk:I

p,(AAAo) < r.

Among others applications of the concept of the sufficientsemiring, w€ mention the following useful approximationlemma.

Lemma L.2.L Let (X,A, p) be a nleo,sure space w'ith asufficient sem'iring C. Then, for any A wi,th p(A) ( oo,there erists a sequence Hn I in A such that p(H") ( oo,H" I H ) A, p(H \ A) : 0, and each Hn 'is a countabledi,sjoi,nt union of elements of the sem'iring C.

Proof: Since C ts a sufficient semiring, for any s ) 0, there

exists H(r)- 3t,,Cj € C,such thatAq HG)andj:7

p(H(e) \ A) ( e. Write Hn: H(1) ) H(+) n . . . ) H(*)Since C is a n-system, it is easy to verify that each Hn ts acountable union of elements of C. By the Proposition I.2.L,Hn is a disjoint countable union of elements of C. Further-more, Hn I H, which obviously has the required proper-ties. n

n


1.3 Lebesgue Integration

In this section, we introduce thearbitrary measure space (X, A, p)properties.

1.3.1 Measurable functions

Lebesgue integral on an

and summarrze its main

(); e R).

Definition 1.3.1 A function f : X ---+ IR zs called mea-surable i,f f-'(I) e A for euery I € /(R).

Note that the collection M(X, IR) of all measurable real-valued functions on X is ciosed under all point-wise algebraicand lattice operations (like addition, multiplication, takingthe positive part, etc.). Moreover, M(X, R) is closed underthe point-wise convergence.

L.3.2 Construction of Lebesgue integral

In defining the integration on (X, A, p), ,t is natural to as-

sume that the integral of lla is equal to p,(A), and that theintegral preserves linear operations. This gives us the firststep in the constructing of the Lebesgue integral.

Definition 1.3.2 Let p,(X) < oo and f e M(X,R) be asimple funct'ion:

n

f (") - t \tn,qu

i,:l

1.3. LEBESGUE INTEGRAT/OAI

Then th,e integral of f is defi,ned by

It is also reasonable to have an integration that allows pass-

ing to uniform limits of integrands on sets of finite measure.So the second step of our construction is following

Definition 1 .3.3 Let p(X)bounded, and {g"}" be a sequence of si,mple funct,ionsconuerging to f uni,formly. Then

ffI f @) p,(d,r) :- Iim I g,@) 1t(d,r).

J n---+6 J

From now on we ailow the Lebesgue integral to take infi-nite values. To avoid minor technical difficulties in the nexttwo steps of our construction of the Lebesgue integral, w€consider non-negative functions.

Definition 1.3.4 Let p(X) < oo and0 < / € M(X,R).Then

where

19

^nI

I f @) t'@r) '- ! \p(At).Iv :_1

X L-T

I tol u@,),-;,* | r.@) p,(d,r)

XX

r*(*):{ r(.) li W}'1")=t'

20 CHAPTER 1. ELEMENTS OF MEASURE THEARY

Now we remote the condition p(X) ( oo, by using an ideasimilar to that one in the last definition.

Definition 1.3.5 Let0 < f e M(X,R). Then

{x"l p@r):- sup

{l r@) p(d*),p(A). *} ,

where

I r wl rr@*) ,: I na,(*) f (r) p,(d,r).

AX

The last step is nothing else than the following technicalagreement:

Definition 1.3.6 Let0 < f e M(X, lR). Tlr,en

I tf"l 1t(d,r):- [ f @) p,(d,r) - [ r@) p(dr),J r' \ /' \ ./ J / \

i,f at least one of integrals 'in the ri,ght-hand si,de i,s fini,te.

The construction of the Lebesgue integration on (X, A, p)is complete.

Remark that the last technical agreement, which allows us

to deal with integrals which are not necessarily finite, cannotbe used for complex-valued functions (or, more generally, forfunctions with values in IR" for n > 2).

1.3. LEBESGUE INTEGRATIOAi 21

In our course) we shall often make of use the collectionLr(X, A, p) of all Lebesgue integrable real-valued func-tions on X (i..., functions / such that f f (") p(dr) exists

and finite). It is obvious that Lr(p) - Lr(X,, A, p) is a vec-

tor space. To simplify notations, we shall often write I f ap

instead ot I f @) p(d")X

Exercise 1.3.1 Show that th,e quoti,ent spa,ce

L,(p) - Lt(p)t {r

. L,(p) , I tftdr,: ,}

'is normed space w.r.t. ll ll r : Lt(p) - R defi,ned by

il t/t il, '- I ,,t du

X

The Lebesgue integral has some important properties thatwe shall often use. We state them without proofs.

1.3.3 Properties of the Lebesgue integral

Theorern 4 (Monotone Convergence Theorem)Let {f"}" g M(X,IR) sofzsfy 0 S f" I f a.e. Then

rfI f dp-lim I f"dp. n

J NJXX

CHAPTER 1. ELEMENTS OF MEASURE THEORY

We also refer to the monotone convergence theorem as toMCT. Among other things, it allows the integration of point-wise convergent series with non-negative terms:

Theorem 5 (Fatou's Lemma) Let {f")n e M+(X, R).Then

Remark that the inequality in the Fatou lemma may bestrict. For example, if (X, A, p) - ((0, 1], t, )) and fr:, . [(0,*1 then liminf f n - 0, but [- f" dp - 1 for all n.

The situation will be changed drarrro{ti.ulty if we suppose asequence to be bounded by a Lebesgue integrable function.

Theorem 6 (Dominated Convergence Theorem)Let {f"}" q M(X,IR), g e Lt(p), and lf"l a s a.e. foralln. If f"(r) -- f (") a.e. then f e Lr(p) and

fIim I lf"- fldp- o. n

n_-* r*

We also refer to the dominated convergence theorem as toDCT.

/1rt- inf f,) d"u <timinf I f,ouJ'n'-nJXX

1.3. LEBESGUE INTEGRATIO.^{ 23

Theorern 7 (Jensen's Inequality) Let g : IR ---+ IR be

a convex funct'ion, 'i.e.:

s(ar+(1 -c-)a)<as(r) +(t -*)g(a)for all r,U e R, 0 1a < 1. Then

,({ r,,) = {gff)d,,

for any f e Lt(p). n

L.3.4 Product of measure spaces (optional)

Let (Xr, At, p) and (X2,, Az, pz) be measure spaces. Forsimplicity, we suppose measures pr , Fz to be finite.

Denote by R the following set

R'.- {At x Az: At e Ar, Az e Ar)of all measurable rectangles in X1 x Xz. Note that Ris a z-system. Denote by O the algebra generated by R.

Define the function pr t Fz on R by

l-4 & Uz(A, * Az) '.- ltL(Ar) ' pz(Az)

and extend it to O. This is possible because each elementsof Q is a disjoint union of finitely many elements of R. Nowwe show that the function h & Fz is countably additive onO. For this purpose it is enough to show that h E Fz tscountably additive on measurable rectangles.

D0 co

,6- - l) o".n:I

Define a functian fn: At ---+ R by

f,(,): { p,(AD

li :ei+

Thus, by the MCT,oooooof

D u' s t'r(B") - t pr(A?)' pr(AD - t I fndt, :n:l n:l ":, i?


LetB"-ATxA|,Ai € At,AT € Azforn- 0, 1,...such that

It is obvious thatoo

Dr"@):t'r@8)n:7

IE-'") dt"- t"@?) p'(Ag): FtE t"(Bo)

Thus Fr I Fz ts countably additive on the algebra O. Anapplication of the Caratheodory theorem gives the measure,\ on the o-algebra o(Ar* Az) which extends ht 1t2. Themeasure .\ will be denoted by the same symbol W g pz.

Definition 1.3.7The n'Leasure space (X, x X2, o(A1 x Az), h g p,2) ,is

called the product of the n'Leasure spaces (Xr, At, p)and (Xr,, Az,, pz).

1.3. LEBESGUE INTEGRAT/OAI 25

Theorem 8 (The F\rbini Theorem)Let f (rr, rz) be a real-ualued Lebesgue ,integrable func-t'ion on X1 x Xz. Then the second and the thi,rd iterated'integrals below make sense and enjoy the formula:

I",*r,f (*',rz)dtttu t": I*,(1.,f (*,,rz)dttz) or, -

L,Ur,f ("'rz)dttr) ou'' n

Theorem 9 (The Toneli Theorem)Let f (*r, rz) be o, real-ualued non-negat,iue measurable

funct'ion on X1 x X2. Then the funct'ions

fz(rr)- t f@r,rz)dpz k ft(*r)- f f@,,,r2)dt-rtJx, Jxr-

are At- and Az-rneasurable, respectiuely, and

I*,,r,f (*'' rz)dpt' & t" : lr,(lr,f (*'' rz)dttz) o" -

L'U"f (*''rz)dtt) o"' n

Example 1.3.1 The non-negatiui,ty cond,ition'in tlr,e Tonel,itheorem 'is essent'ial. Indeed, consider the measure space

CHAPTER 1. ELEMEN"S OF MEASURE THEORY

(N,2(N), )) , where \ i,s the count'ing n'Leasure. Then forthe funct'ion f (r,A) - ?l)"-, . I[1e+1>,>a] 'it h,olds:

oo oo oo oo

I I r@,y)drdy -o+r: I I tn,y)d,yd,r11 11

Remark that i,t is ea,sy to modi,fy the the construct'ion tomake f (*, il cont'inuous on R?.

1.3.5 The Radon - Nikodym theorem and the Riesz rep-resentation theorem

In this subsection we formulate two very useful classical the-orems of the theory of the Lebesgue integration. The firsttheorem deals with a representation of measures throughfunctions and the second one with a representation of posi-tive linear functionals through measures.

Let (X, A, p) be a measure space. A measure u on A is saidto be p-continuous if for every A e A

p(A) - 0 => "(A) - 0.

Theorem 10 (The Radon - Nikodim Theorem)Let (X,A, p) be a o-fini,te n'Leasure space, Iet u be a p"-

cont'inuous n'Leasure. Then there erists a non-negatiue

functi,on f e L,,(lt) such that

,(A):lfdt, (vAeA). n

A

1.3. LEBESGUE INTEGRATIOAI

Let X be a locaily compact metric space. For simplicity,one may think that X g lR'. BV Co(X) we denote thelinear space of all continuous functions X -* IR vanishing oninfinity.

Definition 1.3.8 A mappins d: Co(X) * IR zs calleda) a linear functional i,f d@ft + 0 fz) : ad(f t) + gdjz)for all a,0 € R, ft, fz e Co(X);b) a positive functional i/

f (") > 0 (Vr e x) =+ d(/) > 0

for all f e Co(X).

Theorem 11 (The Riesz Representation Theorem)

For eaery pos'it'iue l'inear funct'ional Q : Cg(X) ---+ IR, thereis a unique fini,te measure LLO on the Borel algebra B(X)such th,at

0(h) - h(") pro@r) (Vh e Co(x)). n

Remark that the measwe p6 in the Riesz representation the-orem is obviously finite. Moreover, it is clear that every finitemeasure LL on B(X), gives a rise to a positive linear func*tional d, on Co(X) by

or@) : I h@) p(dr) (vh € e(x)).X

IX

Chapter 2

Elements of MeasurableDynamics

2.t Examples of Transformations

An abstract semi-dynamical system with discretetime is apair (X,r), where X is aset and r . X ---+ X is amapping. We think of X as the set of states of (X, r) andof r as a law which controls the way how the system (X, r)evolves through time. In the case of an invertible r, thepair (X, r) is called an abstract dynamical system. Ifwe consider a one-parameter semigroup (rr)r>o (one-parameter group ("r)r.o) of transformations of X, thepair (X, (rt)t) is called an abstract continuous semi-dynamical (dynamical) system with the state-set X.In this lecture, we study several elementary examples oftransformations of measure spaces. We require them tobe measurable (not necessarily invertible) transformationsof a measure space (X,D, p). The study of such dynami-cal (semi-dynamical) systems is the subject of the ergodic

CHAPTER 2. ELEMENTS OF MEASURABLE DYNAMICS

theory. Remark that, in many cases, the state-set possesses

a typological structure. This is a very reach in applicationsarea of theory of dynamical systems, which is called topo-logical dynamics. Topological dynamics has been closelyrelated with ergodic theory. The both theories go back toworks of Henri Poincare in the end of 19-th century.

z.L.L Endomorphisms of a measure space

Let (X, A,, p) be a measure space. A mapping r : X ---+ Xis called measurable if r-1(A) e A for every A € A.We usually suppose A to be p-complete (i.e. B g A e Aand p,(A) : 0 implies B e "4), otherwise one may take thecompletion of o-algebr a A.A measurable mapping r : X ---+ X is called nonsingularIf p,(A) - 0 implies p(r-t(A)) - 0. Nonsingular mappingsplay a key role in the concept of a Perro Frobeniusoperator, which will be considered later. We do not needthis concept in the present lecture.

Definition 2.f.L A meo,surable mapp'ing r: X --+ X ,is

called endomorphis rn i,fp(r-'(A)) - p(A) (vA e A).

Remark that an endomorphism r is not necessarily invertibleeven in the case when ,(X) - X and pt(X) < oo (see thedoubling transformation below). The composition of twoendomorphisms is obviously an endomorphism.Our main aim in this lecture is to give several classical ex-amples of endomorphisms.

2.1. EXAMPLES OF "R/NSFORMATIOI{S

2.L.2 Baker transformation

Let X: [0, 1] x [0, 1], It be the Lebesgue measure, and Abethe Lebesgue o-algebra on X. We define a transformationS:X+Xby

A<1;y <7.

To understand the operation of this transformation better,examine the following figure and take into the considerationthe way by which the traditional baker is kneading dough.

31

s(,,r) : { l\:'iq,L,y+}) i2:l l,3l

I ol-

-+'lrrrniffi--The mapping ,S : X ---+ X is called a baker transforma-tion.

Obviously, ,S is measure preserving and invertible. Some-times, the following 3-dimensional version of the bakertransformation:

r(r,A, z) - (S(r,A), z)

is useful. Iterations of the baker transformation are describedby the figure beiow.

I ,l'l-lEJrr sf tttr r-L!-,r---T---; a

32 CHAPTER 2. ELEMENTS OF MEASURABLE DYNAMICS

2.t.3 Rotation transformation

For any a e IR, define the rotation by a to be the trans-formation

Ro : [0, 1) -* 10, 1)

given byR*(*)-(r+*)(mod1).

Clearly, ,R* is invertible and measure preserving. It is also

clear that B* is periodic rff a € a Here we consider theLebesgue measure on 10, 1). If a measurable transformationon (X, A) is given, then a measure p is called invariantmeasure for r lf r is an endomorphism of (X, A, p). Thusthe Lebesgue measure ) is an invariant measure for ,Ro.

Exercise 2.1.1 Let r be an 'inuert'ible measurable trans-

format'ion of (X, A, p). Show that ris an endomorph'ismiff ,-' is an endomorphism.

We usually identify [0, 1) with the unit circle f by

t € f0, 1) * 0(t) - "2rti € f.

It is clear that the rotation transformation -R, becomes theusual rotation Eo("'"tu) :

"2r(t+a)i on l.

The following classical theorem is rather useful.

Theorern L2 (Kronecker) Letr € 10, 1). The sequence

@T,@))Po 'is dense in 10,1) iff a € R \ a

2.1. EXAMPLES OF TRAAiSFORMATIONS

Proof: It is enough to show that the set {RX(r) ' ", > 0}

is dense when a is irrational. Fix a € R \ Q and denotefor simplicity Ro by R. AII the points Eo(") : n, Rt(*),...) R(*)) ...are obviously distinct. By the Bolzano -Weierstrass theorem, there exists a convergent sequence in(R(*))Po. Thus for a given j > r > 0, there exist non-negative integers pHence lRe-q@)-rl < e. Let r - p-q and d: lR'(*)-rl.Then the consecutive terms in the sequenc" (Rt'(r))pn ared-apart of each other. In fact, for I ) 0,

lg(t+t),@)-R,(*)l _ lRt,(R,r)-R,(r)l _ lR,*_rl_ di.

Since +density of the sequence (R(r))Po in [0, 1). n

Although the statement of the last theorem is pretty simple,it will lead us to several important concepts. The first oneis the notion of a minimal map r . X ---+ X on a metricspace X. The minimality of T means just the density of thepositive orbit {r"*}70 for any r € X. The Kroneckertheorem states exactly that all irrationai rotations -R* areminimal.

Among others applications of the rotation transformationEr, is the construction of a subset of 10, 1) that is not Le-besgue measurable. Fix an irrational a. Then the full orbitl, : @2")*?-* consists of distinct points. Moreover, forevery frr, fr2, the orbits lrr,,lrrare either disjoint or coincide.By using the axiom of choice, take in any family of all orbits,which are equal to the same set l, a point r € f. Denote


the collection of all such points by A. Then

lo, 1) - [J r".r€L,

By the construction of A,

lo, 1) - ll R3(^)n€Z

Assume that A is Lebesgue measurable. If ,\(A) - 0 then

1:)([0, 1)) -I)(A) -0.N?L

If)(A) >0then

1:)flo, 1)) -I)(A) :oo.n€Z

So, in both the cases, we get contradictions. Hence A is notLebesgue measurable.

2.L.4 The doubling map

Our next example in study of transformations that is calledthe doubling map on 10, 1) is defined by

r(r)-2r(m,od1) -['*' if o.a"tit

Izr-r if *5"<i.Iterations of r are illustrated on the following figure.

2.1. EXAMPLES OF TRAAISFORMATIONS 35

[l' rl 17

Vl/+"W,-,;\ffi-

An easy investigation of the graph of r on the figure abovegives us that r is a non-invertible endomorphism of [0, 1).

Let D consist of all reals in X : 10, 1) of the form fi, andlet Xs - X \ ,. The numbers in Xs have a unique repre-sentation in binary form as

(ou e {0, 1}).

We call the sequence (o.)|rthe symbolic binary repre-sentation of r.Our doubling map r acts on X0 as the left shift

oo

r(r)-IT @exo)i:7

This representation of r shows among their things the den-sity in [0, 1) of the set of periodic points of r. To see this,

note that the set D of all numbers of the form r - i #,i,:I

oo

r-\-gL-t )r'i,:I

36

k-,L-

I,2,'k

Igg/.-z qz

i,:L

a'i I,nW-r

gives us another dense in [0, 1) set. Obviously, rkn - tr,which justify the density of r-periodic points.

Remark that the doubling map can be constructed on theunit circle instead of the interval 10, 1). This constructionmakes the rotation transformation continuous.

2.2 Recurrence

Definition 2.2.I An endomorph'ism r of a measurespace (X,A, p) is called recurrent i,f for any A e A, al-most eaerA poi,nt of A (i.". all po'ints of A\ltr, /r(lf) - 0)return to A at some future time.

Formally, the recurrence condition is given by the followingformula:

YAeA=tveA(p(lr) -0) &Yr € ,4\ lr h@) > 0 (r"@r e A).

We think of n(r) above as a return time to A.Remark that if the measure pl is infinite, then it may easilyhappened that an endomorphism is not recurrent. For ex-ample, the shift transformation ,(*) - r * 1 on IR. Later

2k(Y+ Y ?n - r2i' ./ '2k+t I "' I

i,:k*1

(1)


..,fr,..., is dense in X - [0, 1). Replacing any

€ D by the (rational) number

n+I)k

Ti,:nk

2.2. RECURRENCE 37

on in Theorem 13 we shall see that this may not happenedwhen p is finite.

Exercise 2.2.L Let r be a recurrent endomorplt,ism of(X, A,, p) and A e A. Show that there erists a null setlf1 such that for any r € A\ m tltere'is an,increas'ingsequence (depending on tr, of course) (*u)?, wi,th

r*itr€A (VieN).

(Ulntt (Jse the set N from (1), consid.er N1 : ULot-*(l[), and, note that,

for eueryr e,4\ Nl c,4\N, r'(")s e a\rur )

2.2.L Conservative endomorphisms

The following observation is an immediate consequence ofDefinition 2.2.1: an endomorphism r of (X,A, p) is recur-rent iff

-0 (VAeA).'(^\C '-r@)) (2)

The following lemma gives a useful characterization of recur-rence.

Lemma 2.2.L An endomorph'ism r on (X,A, p) ,is re-current iff r is conservative (i,.e. for anA g, p(B) > 0,there erists n ) 0 such that p(B ) r-n(B)) > 0).

Proof: Suppose r to be recurrent and let p,(B) > 0. Then

r("\C B),-r@)) - f,("t (""!j,"--,",) ): F("tp,'--,",) -o

Since p@)


n ) 0. Thus r is conservative.

Let r be conservative and let ,4 € A. Denote by B theoo

set A \ U ,-"(A) from formula (2) above. It p,(B)n:1'

then there is ncontradicts the definition of B, since no point of B belongs to,-"(B) C r-n(A) Thus p(B) - 0 and hence r is recurrent.tr

2.2.2 Poincare Recurrence

The following result is one of the first theorems in ergodictheory. It is due to Henry Poincare (1899).

Theorem 13 (Poincare Recurrence) Let(X,A, pt) be

a finite measure space. If , : X ---+ X 'i,s an endomor-ph'ism, then ris recurrent.

Proof: By Lemma 2.2.1, it suffices to show that r is con-servative. Suppose, on the contrary, that p(B)

2.2. RECURRENCE

p(B n r-"(B)) -k>m)0,t'(r-k (B) n '

39

0 for aII n

Therefore

/x \ oo oo

p I U,-"(u) I - t p(,-"(B)) - D p(P)\n:o / n:0 n:0

a contradiction, as p(X) ( oo. Hence

p(B)>0 + ln>0 p,(B)r-n(B)) >0.

-* (B)) - rt(r-^ (r-(r'-^) (s) n B))

F(r-u'-^)(F)nB) _0.

: OO,

2.2.3 Incompressible endomorphisms

Now, we present another characterization of recurrence inthe case of o-finite measure. For this purpose, w€ need thefollowing definition.

Definition 2.2.2 A set C g X 'is called compress-ible for an endomorph'ism r : X ---+ X i,f r-I(C) g Cand p(C \ '-t(C))incompressible if there is no cornpressi,ble set for r.

Theorern L4 Let r be an endomorphi,sm of a o-finitemeasure space (X, A, p) . Then r is recurrent i,ff r is'incompressible.


Proof: Suppose r to be recurrent and let ,-t (C) g Cfor some C € A. Put B - C \ "-'(C). The sequence(r-"(B))Po obviously consists of pairwise disjoint sets.

t-/(c >

C

In particular, ,-"(B) n B - g for all n ) 1. By Lemma2.2.I, p(B) - 0. So r is compressible.

Now let r be incompressible and let A € A. Put

C - U ,-r(A) Then ,-t(C) g C. Obviously,k:0

oo

c\'-'(c)-A\Ur-*(A).k:I

oo

is incompressible, p,(A\ U "k:Ir is recurrent, accordingly to

Asr

that

-r(A)): o, which means

condition (2). n

2.3. ERGODIC TRAAISFORMATIONS

2.3 Ergodic Transformations

Ergodicity is one of most important concepts in our course.It was introduced by George Birkhoff and Paul Smith inT928.

Definition 2.3.1 Let r be a nons'ingular transformat'ionof a rneasure space (X, A, p). Then ris called ergodi c i,feuery r-'inuariant set A e A (i.e. p(r-'(A)LA) - 0) ,is

tri,u'ial i,n the sense that e'itlter p(A) - 0 or p(X \A) - 0.

We are mostly interested in ergodic endomorphisms onfinite measure spaces. However some interesting resultshold true for arbitrary nonsingular transformations.

Theorem 15 Let r be a nons'ingular transformation of ameasure space (X,A, p). Tlr,en r is ergodic i,ff, for euerAmeasurable funct'ion f . X - R,

f o, = f (p-u..) + f : const (t -u..).

Proof: ((-;)' Let r be ergodic, and let / . X -+ IR be ameasurable function satisfying f " r = f (lt-".".). Assumethat / is not a constant (p-*..). Then, for some a € IR,

the sets A - {f < a} and B - {fmeasure. These sets are obviously r-invariant. For example,

,-'(A) - {r: rtr € A} - {f or < a} p-Le' {f < u}: A.

This contradicts to the ergodicity of r. Hence

f - const (p-u.. ).

4l


(( z )) Assume that r is not ergodic. Take a nontrivialA e A that is r-invariant. Then the indicator functionof A is not a constant (p-a.e.). However,

Ila o r :n,-rpa1q-Lt' [o. n

Obvious examples of ergodic endomorphisms appear if themeasure space is trivial.

Exercise 2.3.L Show that the i,dentity transformat,ionon a canon'ical Lebesgue space X 'is ergod'ic i,ff X ,is &s'ingleton.

A less trivial but stiil rather simple situation arises in thecase of baker endomorphisms.

Exercise 2.3.2 Show that the 3-d'imens'ional baker en-domorph'ism 'is not ergodic.

Show that 2-d'imens'ional baker endomorpltism ,is ergod'ic.

Sometimes the question on ergodicity of a given transforma-tion is fairly simple. Here we mention only that the irrationalrotation and the doubling map are both ergodic. However,in certain important cases) the question on ergodicity maybe very deep and difficult.

2.3.L Recurrence and ergodicity

The shift transformation r(n) - n * 1 on Z is ergodic butnot recurrent. Let us study conditions under which a trans-formation is recurrent and ergodic.

set

Ila

2.3. ERGODIC TRAATSFORMATIONS 43

Theorem 16 Let r be an endomorph'ism of a o-finitemeasure space (X, A, p). Then the follow'ing cond'it'ions

en'ists n ) 0

erists n ) 0

are equ'iualent:(1) " 'is recurrent and ergodic;(2) For euery A e A, p(A) > 0,

loo\plx\U,-"(A)l -o'\"Zr/

(3) tf p(A) > 0 and p(B) ) 0, then theresuch that

,-"(A)nBla;(4) tf p(A) > 0 and p,(B)such that

pt-"(A) nB) >0.

Proof: (t) + (Z). Let 1t(A) > 0 and denote B --Since r is recurrent,

oo

Un:1

,-" (A)-

/oo\p l r-'(A) \ U r-"(r-'(A)) I : p(s\ "-'(u)) - 0.

\n:r/Moreover r-1(g) q B, and hence B is r-invariant. By theergodicity of r, the condition p(B)

B - x (r'-u.e ), or r, ( x fl3 "-"(A)) - p(x \ B) : 0\n:r/

(2) =+ (3) Suppose p(A)r-"(A) ) B - a for aII n

44

by (2),

(3) + (4)

t'?-"(A) nC - A \,-"(C) ) B

Let p(A)B)-oo

u?-"(A)n:l- a for all

S f (") t'-Le' f (r("))


oo

X - U ,-"(l) (p- a.e.) and thenn:7

oo

B p-Le' U ("-"(A) n B) _ s.n:I

0 for all nn B) Then p(C)

n ) 0, which contradicts (3).

(4) =+ (1). The recurrence of r follows, by Lemma 2.2.I, if wetake B - Ain (a). Assume that r is not mean ergodic. Thenthere exists a r-invariant (lr-u...) set A such that p,(A) ) 0,

p,(X \ A) > 0. By (+) applied to B - X \ A, there existsn)0suchthat

0 - p(s) - p(An (x\A)) - p?-"(A) n (x\A)) ) 0,

a contradiction. Therefore r is ergodic. n

2.3.2 Spectral properties of endomorphisms

Let r be an endomorphism of a probability space (X, A, p).The endomorphism r induces a linear contraction S on Lr(p),,which is called the Koopman operator corresponding torby

(f e Lr(p))-

2.3. ERGODIC TRAAISFORMATIONS

A number ,\ € C is called an eigenvalue of r if ) is aneigenvalue of S (i.e. there exists 0 + f € Lr(p), whichis calied an eigenfunction or eigenvector correspondingto )). Obviousiy, ) : 1 is always an eigenvalue of r.

Lemma 2.3.L If ), i,s an e'igenualue of r th,en lll - 1.

Proof: Remark that [a o r : \r_r1a; for all A e A. Hence,

by the linear approximation, I g "rdp : I gdp for all

S e Lz(p). Consequently, we obtain that for any f e Lr(p)satisfying S f - f o r : ),f Qra.e.):

which

Exercise 2.3.3 Show that Lf ^

'is e'igenualue, then theset

E()) -{f e Lz(p) ,f o,-^f (p-u")}'is a uector subspace of Lz(p).

Exercise 2.3.4 Let f1, fz b" ei,genfunct'ions correspond-ing to e'igenualues )4 + ),2. Show that f1 and f2 a,re

OfthOgOnAl. (nmt, (Jse the property that I s o r d,pr, : I s d,u for all s e rr|t).)

Lemma 2.3.2 Let T be an endomorph'ism of X.(1) If ,'is ergodic and f is an eigenfunction, then

l,rf ar,: Iv?ordLt: Ivo,r'dt,-

| ,^rrrl' d,t - r^ r I vr'd,r,,

impliesl)l -tn

46

l/l(2)Es


: coristant (p-a.e.);T 'is ergodic i,ff, for each e'igenualue ),, the subspace

- {f € Lz(p) , f o, - ^f jt-u.".)} ,r l-d,imens,ional.

Proof: (1) Let f o r : \f }t-".e.) for some ), l)l - 1

(rf. Lemma 2.3.I). Then

l/l "r-lf"rl -l)/l -l/lThe ergodicity of r impiies that l/l - constant (p-u.. )Moreover, f + 0 (p-a.e.) since / is an eigenfunction, there-fore we may choose an eigenfunction u - #, such that

lrl -1(p-ae)(2) Suppose that r is ergodic and fi o r - \ft jt-u." )From part (1), /r l0 (p,-a.e.). Let f2be another eigenvectorcorresponding to ,\. Then the function g : fzlf, e Lz(p).Since

fzor fzp-d.e.

fr: C'f,

Suppose that dim(E1) - 1 for every eigenvalue ). Con-sider the eigenvalue 1. Thus the only functions / satisfyingf (r(")) - f (r) }t-u ".)

are constant. By Theorem 15, r isergodic. n

By the previous lemma, we may assume that eigenfunctionsof an ergodic r have absolute value 1.

(lr-u...) by

90T-

then g - constant(rf. Theorem 15), say

that dim(81) - 1.

fto, fr -- 9,,

the ergodicity of r(p-u...). This shows

2.3. ERGODIC TRANSFORMATIONS 47

Theorern LT Let r be an ergodic endomorphism of a Le-besgue probability space (X, A,, p). Then the set oo?) ofeigenualues of r is a countable multi,pli,cat'iue subgroupo/S1 -{l€C,l)l -1}Proof: If )r, ,\z € or(r) then fi o r : \th,

We mayfz o , - \zfz for some nonzero h, fz € Lr(p).assume that l/tl - lfrl:1 (p-a.e.). Then

(fr- fr) " r - (/t " r) . (ft o r) - .\r . ),2. (ft. f) 0t-".".).

Thus f, . f, € Lz(p) is an eigenfunction corresponding to,trr ' )2.

Also,1

o r-ft

and f e Lz(p).

This shows that oo?) is a multiplicative semigroup. Noticethat the eigenfunctions corresponding to different eigenval-ues are orthogonal, since

fI h'fzdp-

JXf) o rd,p, -

f2 o r)d"F : I erfr)(XffilaF :Jxf

)rlz I f,' f, dpJx

As Lz(p) ir separable, oo(r) is count-

1 p-d.e. 1

ho, ,\r ft

L,',(f,

implies thatable. n

"r) '(

hLfz

t.

Chapter 3

Ergodic Theorems

3.1 The Birkhoff Ergodic Theorem

The aim of this section is to prove celebrated Birkhoff's in-dividual (: point-wise) ergodic theorem (it will be also re-ferred to as the BlBtheorem) which was obtained by GeorgeDavid Birkhoff in 1931. Nowadays it has enormous amountof applications in different areas of mathematics (probabilitytheory, dynamical systems, analysis, geometry, etc.). We for-mulate also two famous generahzattons of the BlE-theorem.For more advanced individual ergodic theorems we refer toKrengel 15] To avoid unnecessary compiications, we con-sider an endomorphism r : O ---) O of a o-finite measurespace (Q, A,/r) We shall use the following notations:Tf :T,f : f or,

s"f - s,g)f - sTf :A,f : An(T)f - ATf -

n-7Di,:0

Tuf,

*s"g)f49

n-l-l_r--n,/, i,:0

Tnf,

50 CHAPTER 3. ERGODIC THEOREMS

Mf f - max( &f , Szf , . . ., Snf),M"f - max( Arf , Azf , . . ., Anf),M*f - supf,, Anf ,

where / is any real-valued measurable function on O.

3.1.1 Birkhoff's individual ergodic theorem

Theorem 18 (Birkhoff's ergodic theorem) Let r be

an endomorph'ism of (Q,A, p,) and f e Lt - Lt(Q,A, p).

Then Arf u-4",e' t € L1 fo, a" r -,inuariant t(i.". Tf : f o r'-Y"' f) satisfyi,ns

ffI fdp- I fdp'^l

for euery r-'inuariant A e A, p(A) < oo.

The proof of the theorem, which will be given below, is basedon the following maximal ergodic theorem, that is dueto Shizuo Kakutani and Kosaku Yosida (1939) for T gener-ated by endomorphisms, and that is due to Eberhard Hopf(1954) for arbitrary positive contraction 7 in L1.

Theorem 19 (Kakutani -

Yosida -

Hopf) Let Tbe a pos,it,iue contract,ion on L1. Then I f ap > 0, wltere

En

En - {Mf f > o} : {M,f > o}, n € N.

The proof of the Kakutani - Yosida - Hopf theorem whichis given below is due to Adriano Garsia (1965).

f > mf f -r(rwfl)*)Now we integrate this inequality over Er:

I rou. I @rf r _ r(w:ry))dr,-En En

3.1. THE BIRKHOFF ERGODIC THEOREM 51

Proof: Yk: r,...,fr (M:f)* 2 Snf and hence

f +r((M:f)*) > f +Tskf : sr,+tf .

Therefore f > Snf -f (Mf,/)*) for all k - 1,... ,ntI,because it is obvious for k - 1. Passing to suprema? weobtain

: I (Qwf il* - 7(( Mf f)*)) ap -En

: I @: f)*d'1t' - I 'rw:f)*)dp >

cl En

. lw:fl*d'pt- | ,uu:f)*)dp> o.

f) c)

Remark that the last inequality is true, because T > 0 and

llrll < 1 IWe need also the following lemma.

Lemma 3.1.1 (The maximal ergodic inequality)Let r be an endomorph'ism of (Q, A, p), then

ll/llr 2 a. p(M"f > a) ) a- p,(M"f > a)

for all real-ualued f e Lt, a ) 0, and n e N.


This lemma was already known in 1930-th and belongs tomathematical folklore related to the low of large numbers.

Proof: By the maximal ergodic theorem,

fJ U-a)dP>o

{M"f >a}

For any c,,, € {tW"fAnf (r) ) a. This implies Sn(f - a)(r) > 0 and

ll/ll'> I fdpla I 7d,p-a'F(M,f>0).tr{M"f >o} {M"f 2a}

3.L.2 Proof of Birkhoff's individual ergodic theorem

By linearity of T, we may suppose / > 0. Because of

An+tf - (n+ 1) -t f + #(Anf) o r,

the functions

Iu_f - - Iim sup A^f and f' : lim inf A,f

are r-invariant. We shall prove that f"ft : f" l.r-a.e.

The r-invariant set {f" - +*} satisfies

oo oo

{f"-+*}snU{u"f >* (vr>o). (1)m:l n:nl

3.1. THE BIRKHOFF ERGODIC THEOREM 53

By Lemma 3.1.1 we get

ll/llr > .yp,(M"f > t) (Vn e N,7 > 0).

Since {M"fobtain

ItU" - +*) < p,(M*f - +oo) < LL(M*f > j) :Iim p,(M,f > 7) < {'ll/ll, (Vry > o).

n-+@

Hence f"<x p-a.e.

Suppose that for some aB - {ft < CI} n {B < f"). By the r invariance of B, thefunction f ' : (f - 0) .ila has the property:

(f' "'\' lls-,\r P-le' o (Vk > o)'

Hence

lr'ou> I r'd,p- | r'dl,>oB

() r{*,t>o} ?) r{*,f ,>o}

The last inequality is due to Theorem 19. Thus

l ror>p.t@).B

A symmetric argument with f " : (a - f) .fie gives

fI f dp I a. p,(B).

JB

CHAPTER 3. ERGODIC THEOREMS

Together, this contradicts to a < B.

We have shown that A"f converges p-a.e. ifhence if f is arbitrary real or complexLr(Q,A, p).

To prove the last assertion of the theorem? wethat A - O, lr(O) ( oo, and / > 0. For anyM, such that

ll/ - (/ n M,)ll' - ll(/ - M,)* ll' sThen

ffunction of

may assume

e ) 0, take

4,.

I a,r - M,)*dl.t < | uu - Mu)+dlt -Jr, C'

rJU-M,)*dp'ae.o

Thus the sequence (A"f)" is uniformly integrable and henceit converges in ll . ll1-norrrr as it converges p-a.e. to f . Inother words,

I f or- ll t,p A,fll, - Lp llA,fll, - ll/ll, : I f dp nJnn(-) c)

3.1.3 Relatives of the BIE theorem

The BIE theorem has many well known useful generaliza-tions (see, for example, t5]) Here we mention only two ofthem.

3.1. THE BIRKHOFF ERGODIC THEOREM

Probably the first one is the Hopf - Dunford - Schwartztheorem which goes back to Hopf (1954) for 7,7(llo) : [o,and to Dunford - Schwarz (1956) for an arbitrary T.

Instead of (always positive, contractive w.r.t. L1 - norm,and satisfying ?(llsl) - Ilcr) operator T in Lr : Lr(Q,A, p)generated by an endomorphism, we consider now an arbi*trary positive contraction in L1 which is also a contrac-tion w.r.t. ,L--norm. Such operators are referred to as

LrL*-contractions.

Theorern 2O (E. Hopf, N. Dunford, J.T. Schwarz)Let (Q, A, p) be a probab'ili,st'ic space and let T be a pos'i-ti,ue L1-L*-contract'ion on L1.conuerge p,-a.e. for all f e Lt.

Then the auerages An(f) f

Remark that the assumption on 7 to be an LyL--contractionis not very restrictive, although it forces 7 to be an Le-contraction for all p,7 < p ( oo.

Another classical generalization of the BlE-theorem is dueto Rafael Chacon and Donald Ornstein (1960). It deals withpl-a.e.-convergence of the ratio S"(T)f lS"g)g for an arbi-trary positive contraction 7 rn Ly Recall that we denote by

n-IS"g)/ the sum D fn f

i--0

n

56

Theorern 2Lcontraction 'in

CHAPTER 3. ERGODIC THEOREMS

(Chacon - Ornstein) Let TL1. Then for any f € Lt, g €

bea, theLI

ratio S"g)f lS"g)S conaerges p"-a.e. on the setoo

{s*Q)g > o} ,- {, € o ' it r"d(") > o}

n:0to a fini,te li,mit. nNote that the BIE theorem becomes an immediate conse-quence of the Chacon - Ornstein theorem, if we put g - IIs;.

3.2 von Neumannts Mean Ergodic Theorem

The mean ergodic theorem of von Neumann deals with normconvergence of averages instead of a.e.-convergence. Althoughthe Birkhoff's paper 12] appeared a bit earlier than von Neu-mann's paper lB] (both papers are published in 1931-32), thevon Neumann's theorem was proved first and was known toBirkhoff. Nowadays the von Neumann mean ergodic the-orem is known as a rather general resuit about norm con-vergence of averages of operators in Banach spaces. In thislecture, we shall prove this theorem for the case of contrac-tions in a Hilbert space, ffi John von Neumann did. Then wemention only some classical generalizations of this theoremand refer the reader to {5] for more information on this.

3.2.I Mean ergodic theorem of von Neumann

First of all, we remind that Ly(Q,A,, p) is a Hilbert space.Instead of dealing with this concrete space, we consider ar-

3.2. VOAI I\IEUMAAI I'S MEAN ERGODIC THEOREM

bitrary Hilbert spaces.

Theorern 22 (von Neumannts mean ergodictheorem) Let T be a contract'ion 'in a Hi,lbert spo,ce"11, and P the project'ion onto the fired space Fix(?) -{S e'11 : Tg : g}, then A"g)f conuerges ,in norm toPf for all f e '11.

Remark that in the case? when 11 - Lr(Q, A, p),/r(O) - 1, and T is generated by an endomorphism of(Q, A,p), the l\ME-theorem follows directly from the BIE-theorem by using the Lebesgue dominated convergence the-orem and the density of L*(Q, A,/r) in Lr(Q,A, p).

Our aim now is to prove the general version of the NME-theorem. Its proof is much simpler than the proof of theBlE-theorem. In order to give this proof (which follows to[5, p.a]), *. shall use the following lemmas.

Lemma 3.2.L LetT be a contract'ion,in a H,ilbert spaceHandf e7{,thenle Fix(T) i,trf €Fix(T-).

Proof:

f -rf + ll/ll' - (f ,rf) + ll fll'- (r.f ,f) +llr. f - fll' : llr. fll' + ll/ll' - 2(r. f, f) :

llT. fll' - ll/ll' ( o =+ T. f - f .

The inverse implication

T.f-f+f-Tfis also clear. n

5t


Lemma 3.2.2 Let U be a family of contractions in aHi,lbert space 71, then the orthogonal complement of the

fired spa,ceFix(U) - {f e Tt:Tf - f Yf e U}'is equalto the closure o/ span{ h - Th : h e H,T e U}.

Proof: Denote by

F - Fix(U), lf : span{h - Th : h e'11,T eU}andwrite f -LY if /isorthogonaltoeachelement ofY g11.

f LIv e (f ,g - I)h):0 Vh €H,,T eU e(T.f - f ,h):0 Vh €H,T eU +

T.f - f e f € Fix(Z): F,

where the last equivalence is due to Lemma 3.2.L Thencl(,nf) : FL what is required. nProof of the NME-theorem: Since

llA"g)(h - rh)ll -n-tllh - Th + Th - T2h + ... + Tn-7h - T"hll -

n-tllh - T"hll < 2n-tllhll - o,

by the approximation

llA"(r)f - Pf ll - llA"g)f - oll * o

for all / e cl(,nrt). It is obvious that

llA"Q)g - Psll - llg - ell - o

for all 9 € Fix(T) By Lemma3.2.2 applied toU - {7}, weobtain Jl - Fix(T) O cl(..nf). Hence llA"Q)u - Pull - 0

for all u e 7{, what is required. n

3.2. VON I\IEUMAAIN'S MEAN ERGODIC THEOREM

3.2.2 Relatives of the NME-theorem

We consider now the following rather general version of thevon Neumann mean ergodic theorem, which is due to W.Eberlein (1949), but the main assertions emerged alreadywith the works of F. Riesz (1938), K. Yosida (1938), and S.

Kakutani (1938).

Theorern 23 (Mean ergodic theorem)Let T be a l'inear bounded operatorin a Banach space X .

supposeT to be cesd,ro bounded (i,.e. rg llA"g)ll < *).For ana r € X sati,sfyi,rg hylln-If"rll - 0 and ana

a e X , the follow'ing assertions are equ'iualent:(t) A € Fix(?) and A e co {r,Tr,T'*,. . .h(ti) y - lim A"(T)r;

n(iii,) y is a weak cluster po'int of (A"Q)")T:r. n

Corollary 3.2.I If T 'is a power bounded linear operator(i.".sup llT"ll < *) 'in a refl,eriue Banach space (for er-

n)0ample, all spaces Lo are refl,eriue when 1 < p < oo) thenT i,s mean ergodic (i,.e. the auerages A"(T)n conaergein norm for all r € X). n

Corollary 3.2.2 AnA LrL*-contraction on Ly ouer aprobabilisti,c sp&ce is mean ergodic. n

A slight generalizalion of this corollary is the following one.

59


Corollary 3.2.3 AnA linear bounded operator on L1 suchthat the set {A"g)r}n2t is almost order bounded (i.r.VsBh {u e Lr: ll"ll < 1}) is mean ergod'ic. n

The following very useful criterion of mean ergodicity is dueto Robert Sine lI2)

Theorern 24 Let T be a Cesd,ro bounded operator sat-i,sfyi,ng hmn-rTny - 0 for all r € X. Then T ,is n'Lean

nergod'ic i,ff Fix(T) separafes Fix(T.). u

Roughly speaking, the Sine theorem says that 7 is meanergodic ltr T has at least so many invariant vectors as 7*has.

3.3 Applications of Ergodic Theorems

3.3.1 A criterion of ergodicity of endomorphisms

We prove now the following useful characterization of ergod-icity.

Theorern 25 Let r be an endomorphi,sm of a probabi,li,tymeasure space (X, A, p). Then ris ergod'ic i,ff for allA,B € A,

n-7

L p(r-* (t) n B) - u@) p(B)k:0

1hm

n---+n Tl,

3.3. APPLICATIOAIS OF ERGODIC THEOREMS 61

Proof: Suppose that r is ergodic and \et A, B e -4. ThenIla € Lt(ti and, by the Birkhoff theorem,

Hence

I ,-7

Jg ;Dno(,r(")) - p(A) (r,-u. )k:0

t n-7

jg; f (lla o ,k) .iln : p(A) .ila (t'-a e ).k:0

Since

I'n-7 I

l; t(r.q " ,\ 'nBl < r 0'-a e )l*:o I

for all n ) 0, the Lebesgue dominated convergence theoremgives

r31 , r3:l fJ*;: t'('-*(A) n B):J*;:

lo-r,A)nBdt'-n<n-I^an_7

I iim 1 ; il,-n6y.ns d,p" - / ^Lt* * tfo Aork).ns d,trt -.l ;:; n .4fr,-n67'nn dtt:

IJ1g *D(raork)'116k k:0 k -k:0

fJ P'(A) ' iln dp - P(A) ' p(B).X

For the converse) assume A to be r-invariant. TakingB - L, obtain

I ,-Ip(A).p(A):

Jgg ;Dp(,-k(A) n A):k:0


1 n-7

- -ig;I p(A) - p(A)n___+@ , ^k:O

Thus p(A) - p(A)2, which means p(A) - 1 or p(A) - 0.

Hence r is ergodic. n

Remark that this theorem can also be proven by using thevon Neumann ergodic theorem.

3.3.2 LJniform convergence of Cesiro averages

Let r : O ---+ O be a continuoustransformation of. a compactmetric space f).

Theorern 26 For any cont'inuous functi,on f e C({l) the

following condit'ions are equiualent:

a) The funct'ions A'nf -n-I

*f f orn,(n> 1) e C(O) a,rek:0

equi- continuous.b) The sequenc" (A;f)" conuerges uniformly on Q.c) The sequenc" (A'"f)" conuerges weaklg i,n the Banachspa,ce C(O).d) The sequenc" (A;f)" conuerges po'int-w'ise on Q.

Proof: a) + b), By Theorem 23, it is enough to showthat the sequence (A'"f)" possesses a weakly convergent sub-sequence. But accordingly to the Arzela - Ascoli theorem(A;f )" has even uniformly convergent subsequences.

3.3. APPLICATIOAIS OF ERGODIC THEOREMS

b) -+ o), It is obvious, since every uniformly convergentsequence of continuous functions on a compact metric spaceis equi-continuous.Implications b) + c) + d) are trivial because the uniformconvergence is stronger than the weak one and the weak con-vergence implies the point-wise convergence.d) + c): Let A[f converge point-wise to a function /s.Remark that it is enough to prove that for any positive linearfunctional m: C(O) ' IR,

J51 m(fo - A'"f) - 0. Accord-

ingly to the Riesz representation theorem (see Theorem 11),there exists a measure p on the Borel algebra B@) satisfying

m(g): lgdtr (vge c(o)).CI

The sequenc "

(A;f ),, is obviously bounded. By the Lebesguedominated convergence theorem, ;* d(f, -

AT,f) dp - 0.

Hence lim m(fs - AT,f) : 0, what is required.rr---+oo

c) =+ b), It follows from Theorem 23. Z

3.3.3 Uniquely ergodic stochastic operators in C(C))

Let O be a compact metric space. A positive operator S inthe Banach space C(0) is called a stochastic operator ifSII : I[, where I[ - IIe (cf. Definition 4.1.1). For example,the operator Sf - f o, in Theorem 26 is a stochasticoperator. By the Riesz representation theorem , the dualof C(O) , M - C(O)*, is the space of signed measures on


B(A). The dual operator S* to S is defined by

(f ,S.p| - (Sf ,p) (V/ e C(o) ,F € M).

Exercise 3.3.L Show that, for any stochast,ic operator Si,n C (Q, there erists at least one invariant probabilitymeasure p € M (that is S* p, : tt). (nln ' rake aBanach lirnit tr

and a po'i:nt a.'e f). Define a pos'i.ti,ue li,near functional p, on C(A) by p(f) : L((S"f (u))n.).

Hence p,€ M: C.(C)). Show that S* lr: lt, lt ) 0, p(lt) : 1.)

Definition 3.3.L A stochast'ic operator S i,nC(O) ,is calleduniquely ergo dic i,f S has only one ,inuariant proba-bi,li,ty n'Leasure.

Theorern 27 AnA uniquely ergod'ic stochast,ic operatorS 'in C(O) 'is mean ergod'ic.

Proof: Let0lu e M, S*u - u. Wemayassume u)0.Since the norm on M is additive, llS-ll - ll,Sll - 1, and

S*u+)S*u-u + S*u+)ut,we obtain that S*u+ : ut > 0. Thus (ll, r*) ) 0, but II isa fixed vector of ,S. By Theorem 24., S is mean ergodic. n

The following theorem is a relative of Theorem 15

Theorern 28 A meo,n ergod'ic stochast'ic operator S 'isn-l

un'iquely ergodi,c i,ff A*f - * D Sk f conaerges un'iformlylc:0

to a constant for each f e C @) .

3.3. APPLICATIOAIS OF ERGODIC THEOREMS 65

Proof: Denote the norm limit of Aflf for f € C(O)bV f Suppose that f - const for all / € C(O). Take,S*-invariant probabilities u1 afid u2, then

(f ,rr) - (f ,li- (Af ).r') - (J5g Al,f ,r') :(f ,rr) - (f ,rr) : (f ,rr) (V/ e C(O)).

Hence u\ : Uz.

Assume now that f (rr) I f @r) Take the Dirac measures5,, and 6,r. The sequences (Af).d,, and (AE).6,2 convergein c,,r*- topology on M : C* (O) to ,S*-invariant probabilities(1 and (2, which are different because of

(/, (r) : lim (f ,(Af,)*d,,) : lim (,af f ,6,r) :??--'oo n--+oo

(f ,6,,) - f(r) I f@r) - (f ,6,,): ff,Cr). tr

66 CHAP'I:ER 3. ERGODIC THEOREMS

Chapter 4

Related Topics

4.L FYobenius - Perron Operators

4.1.1 Stochastic operators on C(f))

First of all we remind the following concept that appearedalready in subsection 3.3.3.

Definition 4.L.L Let F(O) be a uector space of compler-aalued funct'ions on a, set O. A I'inear operatorT : F(O) -+ lr(O) 'is called o stochastic operatorufT i,s positive, that i,s

[(vte o)/(r) >0] + f(vte ox"/)(t) >01,

and preserues tlte funct'ion [o.

We are interested mostly in stochastic operators in the case

when O possesses a structure of compact metric space andF(O) - C(0), the space of all continuous complex-valuedfunctions on O.

67

CHAPTER 4. RELATED TOPICS

In the simplest non-trivial case O - {rt,u)2t ...,an} is dis-crete and finite. Then any linear operator T on C(O) -.C"can be uniquely identified with an n x n matrix (ti)i-T',:j=iThis matrix is called stochastic if it represents a stochasticoperator on C'.

Exercise 4.L.L Show that a matri, (t,)ijT'l=? is stochas-ti,c i,ff all 'its entries t,ii are nonnegat'iue reals sati,sfyi,ngthe property

n

Itur:1?J-L

for any'i: I,2,...,fr.

Exercise 4.I.2 Show tltat the product of anA two stochas-tic matrices is a stochast'ic matrir. Is i,t true for arbitrarystochast'ic operat ors ?

Exercise 4.L.3 Show that the set of all stochastic opera-tors on F(Q), whereA + 0, is o, non-ernptA conr)er subsetof the space r(F(O)) of all linear operators on F(A).

4.1. FROBENIUS PERROAI OPERATORS

4.L.2 Markov operators in L{p)

Definition 4.1.2 Any positi,ue I'inear operatorT in L{p)such that llT. f llr - ll/llt is called a Markov operator.

Exercise 4.t.4 Sltow that the set of all Markou opera-tors on L1(p,) 'is non-empty, conlefr, and closed ,in theoperator norn'L of L(L1(p))).

Exercise 4.L.5 Show that the dual to a Markou operator'is a stochastic operator.

A matri" (tu)i\?j=? is called Markov matrix if it repre-sents a Markov operator on the finite discrete measure space(n,P(n), )), where

^({k}) - 7ln for all k:0,1,... ,fr - I,

a counting probabilistic measure on n - {0, 1,...,fr-1)

69

Exercise 4.L.6 Show that a matri, (t,.)ijT'l=T ,is Markouiff all its entries t,ii are non-negat'iue reals sati,sfging theproperty

n

Irnr-1frfor any j :1,2,...,fr.


4.L.3 Frobenius - Perron operators

Let r be a measurable nonsingular transformation of a mea-sure space (X, A, p). We do not impose any restriction onr and on X apart from the common assumption that p iso-finite.Letfe e following function on A

fdp (AeA).

Clearly, uy rs a signed measure on A. Moreover, uy is fr-nite and pr-continuous. By the Radon - Nikodim theorem(Theorem 10), there exists a unique p.f € Lt(p) such that

,r(A) : I pf dLL (VA e A).A

Definition 4.L.3 Let r be a nons,ingular transformat,ionon a o -fini,te n'Leasure space (X , A, p) . Then the uniqueoperator P . Lr(p) -- Lt(p) defi,ned by the equat,ion

Irrdt,- tfdp (vAeA)JJA ,-r(A)

'is called theErobenius - Perron operator correspond-'ing to r.

Exercise 4.I.7 Show that P - P, is a Markou operator.Sltow that for euery two nons'ingular transformations 11,

12 the followi,ng Pr2or1 - Prz o Pn holds.

Lt (t). Consider th

,f(A) - I,_r(A)

4.1. FROBENIUS - PERROAI OPERATORS 7I

Gi,ue an eranxple oJ two Froben'ius - Perron operators Pr,and P,, such that *(P"r+ P,r) is not a Frobenius - Perronoperator.

Although Definition 4.7.3 is highly nonconstructive becauseof the Radon - Nikodim theorem, in some cases explicit for-mulas for Py are available.

For example, lf X : 10, 1] with the Lebesgue measure, then

r

P,f(,) : * I r,t or: * I f dt,.

o r-1([o,r])

This formula allows us to get explicit form of P, If r is apiece-wise diffeomorphism. Let us consider the simplest case

of increasing on [0, 1] function r that is a diffeomorphism.Then our last formula becomes

- f (,-'(,))*V-'(,)lThe formula makes sense if we restrict P on a subspace ofL,,(lt) like ClO, 1] for elements f of which the expression

f (r-t(r)) is well defined.

Remark also that certain multi-dimensional analogues of thisformula can be found.

P,f (*): * j *, dr,- *,' l'' , o, -b ,-i(o)

Exercise 4.I.9 Letrtic map) "(") - 4r(I

72 CHAPTER 4. RELATED TOPICS

Exercise 4.L.8 Let X - fiR wi,th tlte Lebesgue measureand ,(r) - exp(r). Show that P,f (r) - L,f (t"r) for a

cont'inuous f e Lr (R).

P"f (r) -

be tlr.e quadratic map (or logis-- r) on 10,71. Show that

(rF) + r(*F)4\tr- r

(Hin ' Show that

'-'co,"l) :

[o, tF1, l-F, r]

and difierentiate the formula which definer e.)

Exercise 4.1.10 Fi,nd the erplicit form of tlte Frobe-n'ius - Perron operator P, correspond,ing to the followi,ngtransformat'ions:a) r : IR+ -* IR1, ,@) - 12;

b) " : [0, 1] - 10, 11, r(r) - sinrntc) r: IR -+ R, r(r) - r mod(1).

Many other interesting information on explicit representa-tions of Frobenius - Perron operators can be found in Lasota

- Mackey book [7].

4.L.4 Koopmanoperators

Let r be a nonsingular transformation on (X, A, p).

4.1. FROBENIUS _ PERROI{ OPERATORS 73

Definitiort 4.L.4 The operator (J, : L*(p) ---+ L*(p)defined by

U,f (*) - f (r(*)) (p-u. )

for euery f e L*(p) 'is called the Koopman operatorcorrespondi,ng to r.

Notice that the usingof p,-a.e.-option in this definition causesminor problems and can be easily avoided if we define LI, onindicator functions by

Urn,q, - [.A o T : I["-r1a;

and approach any f € L*(p) by simple functions inll . ll*-norm.

Theorern 29 The Koopman operatoris adjo,int to theFroben'ius - Perron operator that is

(P"f , s) - (f ,U,g) (1)

for all f e Lt(p), g e L*(p).

Proof: Both operators Pr, (J, are linear contractions. So

it is enough to prove the equation (1) for indicator functions.Let g - IIA and / e Lt(p). Then

(P,f ,ra) : lfr,n. (rra) dt, - I ,,, dp -XA

- I f dp: I f .il,-,(a)dtt,: I f . $aor)dt, -"-1(A) x x


U,(ni dp - (f ,U,na) .

4.2 Mixing

Throughout this section, r is supposed to be an endon'Lor-ph'ism of a probab'ili,ty space X : (X, A, p).

Accordingly to Theorem 25, Lhe ergodicity of r is equivalentto the condition:

, n-|l-_li* =)

^.p(r-r(A)n B): p(A)p(B) (VA, B e A). (1)n-+oo rL u

k:0

4.2.L Mixing endomorphisms

The conditio" (1) is obviously weaker than the following con-dition:

Ii- p(r-k(A) n B) - p(A)p(B) (VA,B e A). (z)rL---+6

Definition 4.2.L An endomorph,ismr satisfy,ing the con-di,ti,on (2) 'is called mixing.

It is worth to remark that the concept of mixing general-izes asymptotically, in a certain sense, the basic conceptof the probability theory, the independence of eventsA,,B € A, that is p(r0(A) n B): tt(A)B) - p(A)p(B)

:lrX

4.2. MIXING

4.2.2 Weakly mixing endomorphisms

The mixing of r means that, for aII A, B € A, the se-

quence (tt(r-r(A) )B))Po of reals converges to p(A)p@)in the usual sense. The ergodicity of T means that, for allA, B e A, the sequence (p(r-r(A)nB))Po ts Cesd,ro con-uergent Lo p(A) p(B). Another type of convergence of realsequences, the strong Cesd,ro conuergence) motivates thefollowing class of endomorphisms.

Definition 4.2.2 An endomorph'ism ris called weaklymixing z/

75

rim lt lr?-r(A) n B) - t@)rru;)l - orr---+oo ,,

k:0

for all A, B e A.

(3)

It can be shown that there are weakly mixing endomorphismswhich are not mixing and ergodic endomorphisms which arenot weakly mixing. Thus we have the following proper im-plications for endomorphisms of a probability space:

mixing =* weakly mixing + ergodic =+ recurrent.For more information on these implications the reader is re-ferred to bookr [3],[7], and [11]

4.2.3 Exact endomorphisms

Another important class of endomorphisms that is consid-ered in this section is given by the following definition.

76

Definition 4.2.3 An,(A)eAforallAeA

Iim p,(rr (l)) -n---+6


endomorphism ris called exact 'if

1 (VA e A, p(A) >

such that

0) (4)

It is obvious that an invertible endomorphism of a non-trivialprobability space cannot be exact.

4.2.4 Ergodicity, mixing and exactness in terms of corre-spondent FYobenius - Perron operators

Recall that an element f of Lr(p) is called a densityfdenoted bv D(p) - D(p).

Theorem 30 Let P, be the Froben'ius - Perron operatorcorresponding to an endomorphism r. Then(1) PiU) conaerges 'in ll . ll yTlorrrl tu n y for all f e Diff , 'is eract;(2) Piff) weakly conuerges to ny for all f e D iff r ,is

m'iring;n-I

(3) lim * D Pi(fl conuerges,in ll .llt -norn'L tofiy for all??-+oo '" k:0

f e D iffr'is ergod'ic.

Proof: We shall prove only that the strong convergenceof Pi ff ) to I[; implies exactness of r and that the weakconvergence of Piff) to II; implies mixing. For other, more

ifis

4.2. MIXING 77

delicate, parts of the proof, we refer to the Lasota - Mackeybook l7l.

Let ll Pif - Il;llr -- 0 for every f e D.Assume p(A) > 0.

Take the density f a - (p(A))-t.lla and define the sequencea, by en : llPi f o - IIx llr. Then

t?"(A)) - tnyd"1t"- tPifa-(pifa-rrx)l d,pt>JJ,"(A) ,"(A)

t pifad*r- t tp?r^-rr;ldp> t pifad,-en:J J trrrA

J,"(A) ,"(A) ,"(A)

I fadp-ena I fadp-en-r-en."_n(rn(A)) A

Since en i 0, we obtain J* p?"(A)) - 1 which means

exactness of r.Now assume the weak convergence of Pi f to

I I / ll r .llx

, where0 + f e f{ 0t). Let A, B e ,4, then

lim 1t(r-r(l)n B) - iim t np d,trt -7?-+oo "-* r-!@)

,,- rim I Piil" dp - tim (pit[s. Ila) -*'* to n'---+oo

- (llnsllr,lra) - p(B)(rr, ra) : p(A) . p(B).Thus the condition (2) is satisfied and hen ce r is mixing. tr

Bibliography

[1] K.B. Athreya, S.N. Lahiri, Measure theory and probabi,li,ty theory, Springer, 2006.

[2] G.D. Birkhoff, Proof of the ergodi,ctheorem Proc. Nat. Acad. Sci. USA, 17 (1931),656 560.

[3] I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, Ergodi,c theory, Springer-Verlag, 1982.

[4] A.N. Kolmogorov, S.V. Fomin, Introductory real analysis. Reu,ised Engli,sh edi.ti,on,Prentice-Hall, 1970.

[5] U. Krengel, Ergodic Theorems, Walter de Gruyter, 1985.

[6] L.B. Koralov, Ya.G. Sinai, Theory of probability and random processes. Secondedi,t'ion, Springer, 2007.

[7] A. Lasota, M.C. Mackey, Chaos, fractals, and noi,se. Stochastic aspects of dynami,cs.S econd ed'i,tion, Springer-Verlag, 1994.

[8] J. von Neumann, Proof of the quasi,-ergodic hypothesis Proc. Nat. Acad. Sci. USA,18 (1932), 31-38.

[9] H.L. Royden, Real analysis, Macmillan, L988.

[10] W. Rudin, Real and compler analysis. Thi,rd edi,tion, McGraw-HilI, 1987.

[11] C.E. Silva, Inui,tation to ergodi,c theory, American Mathematical Society, 2008.

[12] R. Sine, ,4 mean ergodic theorem, Proc. Amer. Math. Soc. 24 (1970), 438-439.

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