Lecture Notes on Ergodic Theory, Lecture notes of Mathematics

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Ergodic Theory

(Lecture Notes)

Joan Andreu L·zaro CamÌ

Department of Mathematics

Imperial College London

January 2010

Contents

• 1 Introduction
• 2 Measure Theory
  • 2.1 Motivation: Positive measures and Cantor sets
  • 2.2 Measures and -algebras
    • 2.2.1 Measures on R
    • 2.2.2 Examples
  • 2.3 Integration
    • 2.3.1 Properties of the Lebesgue integral
• 3 Invariant measures
  • 3.1 Invariant measures: deÖnitions and examples
    • 3.1.1 Examples
  • 3.2 PoincarÈís recurrence Theorem
  • 3.3 Invariant measures for continuous maps
• 4 Birkho§ís Ergodic Theorem
  • 4.1 Ergodic transformations.
  • 4.2 Conditional Expectation
    • 4.2.1 Properties of the conditional expectation
  • 4.3 Birkho§ís Ergodic Theorem
  • 4.4 Structure of the set of invariant measures
• 5 Circle rotations
  • 5.1 Irrational case
• 6 Central Limit Theorem
  • 6.1 Mixing maps
  • 6.2 Central Limit Theorem

Lorenz in 1963 as an extremely simpliÖed model of the Navier-Stokes equa- tions for a áuid áow: 8 < :

x _ 1 =  (x 2 x 1 ) x_ 2 = x 1 ( x 3 ) x 2 x _ 3 = x 1 x 2 x 3 ;

where  is the Prandlt number and  the Rayleigh number. Usually  = 10, = 8= 3 , and  varies. However, for  = 28, the systems exhibits a chaotic behaviour. This is a very good example of a relatively simple ODE which is quite intractable from many angles. It does not admit any explicit analytic solutions; the topology is extremely complicated with in- Önitely many periodic solutions which are knotted in many di§erent ways (there are studies of the structure of the periodic solutions of Lorenzís equa- tions from the point of view of knot theory); on the other hand, numerical integration has very limited use since nearby solutions diverge very quickly.

Using classical methods, one can prove that the solutions of Lorenzís equations, eventually, end up in some bounded region U  R^3. This sim- pliÖes our approach signiÖcantly since it means that it is su¢ cient to con- centrate on the solutions inside U. A combination of results obtained over almost 40 years by several di§erent mathematicians can be formulated in the following theorem which can be thought of essentially as a statement in ergodic theory. We give here a precise but slightly informal statement as some of the terms will be deÖned more precisely later on these notes.

Theorem 2 For every ball B  R^3 , there exists a ìprobabilityî p (B) 2 [0; 1] such that, for ìalmost everyî initial condition x 0 2 R^3 , we have

lim T!

T

Z T

0

(^1) B (xt) dt = p (B) ; (1.2)

where xt is the solution of (1.1) with initial condition x 0.

First of all, recall that (^1) B is the characteristic function of the set B deÖned by

(^1) B (x) =

1 if x 2 B; 0 if x = 2 B:

The integral

R T

0 1 B^ (xt)^ dt^ is simply the amount of time that the solution^ xt spends inside the ball B between time 0 and time T , and T ^1

R T

0 1 B^ (xt)^ dt is therefore the proportion of time that the solution spends in B from t = 0 to T. Theorem 2 makes two highly non trivial assertions:

1. that the proportion T ^1

R T

0 1 B^ (xt)^ dt^ converges as^ T^! 1;

2. that this limit is independent of the initial condition x 0.

There is no a priori reason why the limit (1.2) should exist. But perhaps the most remarkable fact is that the limit is the same for almost all initial conditions (the concept almost all will be made precise later). This says that the asymptotic time averages of the solution xt with initial condition x 0 are actually independent of this initial condition. Therefore, independently of the initial condition, the proportion of time that the systems spends on B is P (B). In other words, there exists a way of measuring the balls B such that the measure P (B) gives us information on the amount of time that the system, on average, spends on B. Theorem 2 is just a particular case of the more general Birko§ís Ergodic Theorem which we will state and prove in Chapter 4. The moral of the story is that even though Lorenzís equations are dif- Öcult to describe from an analytic, numerical, or topological point of view, they are very well behaved from a probabilistic point of view. The tools and methods of probability theory are therefore very well suited to study and understand these equations and other similar dynamical systems. This is essentially the point of view on ergodic theory that we will take in these lectures. Since this is an introductory course, we will focus on the simplest examples of dynamical systems for which there is already an extremely rich and interesting theory, which are one-dimensional maps of the interval or the circle. However, the ideas and methods which we will present often apply in much more general situations and usually form the conceptual founda- tion for analogous results in higher dimensions. Indeed, results about inter- val maps are applied directly to higher dimensional systems. For example, Lorenzís equations can be studied taking a cross section for the áow and using PoincarÈís Örst return map, which essentially reduces the system to a one dimensional map.

2.1 Motivation: Positive measures and Cantor sets 6

to talk about C as having any length. Nevertheless, the total length of the intervals removed is

P

i 0 ri^ <^1 so it would make sense to say that the size of C is 1

P

i 0 ri. Measure theory formalizes this notion in a rigorous way and makes it possible to assign a size to sets such as C.

Remark 3 If

P

i 0 ri^ = 1^ is exactly^1 , then^ C^ is an example of a non- countable set of zero Lebesgue measure.

Non-measurable sets

The example above shows that it is desirable to generalise the notion of length so that we can apply it to measure more complicated subsets which are not intervals. In particular, we would like to say that the Cantor set deÖned above has positive measure. It turns out that, in general, it is not possible to deÖne a measure in a consistent way on all possible subsets. In 1924 Banach and Tarski showed that it is possible to divide the unit ball in 3-dimensional space into 5 parts and re-assemble these parts to form two unit balls, thus apparently doubling the volume of the original set. This implies that it is impossible to consistently assign a well deÖned volume to any subset in an additive way. See a very interesting discussion on wikipedia on this point (Banach-Tarski paradox). Consider the following simpler example. Let S^1 be the unit circle and let f : S^1! S^1 be an irrational circle rotation. We will see that, in this case, every orbit is dense in S^1 (Theorem 27) Let A  S^1 be a set containing exactly one point from each orbit. Suppose that we have deÖned a general notion of a measure m on S^1 that generalises the notion of length of an interval so that the measure m (A) has a meaning. In particular, in order to be well-deÖned, such a measure will be translation invariant in the sense that the measure of a set cannot be changed by simply translating this set. Therefore, since a circle rotation f is just a translation, we have m (f n^ (A)) = m (A) for every n 2 Z, where f n^ := f  n^ times: : :  f. Moreover, since A contains only one single point from each orbit and all points on a given orbit are distinct, we have f n^ (A)\f m^ (A) = ; if n 6 = m. Consequently,

m

f n^ (A)

[

f m^ (A)

= m (f n^ (A)) + m (f m^ (A)) :

Therefore,

1 = m

S^1

= m

[ 1

n= f n^ (A)

X^1

n=

m (f n^ (A)) =

X^1

n=

m (A)

2.2 Measures and -algebras 7

which is clearly impossible as the right hand side is zero if m (A) = 0 or inÖnity if m (A) > 0. In order to overcome this di¢ culty, one has to restrict the family of subsets which can be assigned a length consistently. This subsets will be called measurable sets and the family a -algebra.

Remark 4 The previous counterexample depends on the Axiom of Choice to ensure that the set constructed by choosing a single point from each of an uncountable family of subsets exists.

2.2 Measures and -algebras

Let X be a set and A a collection of (not necessarily disjoint) subsets of X.

DeÖnition 5 We say that A is an algebra (of subsets of X) if

1. ; 2 A and X 2 A,
2. A 2 A implies Ac^ 2 A,
3. for any Önite collection A 1 ; :::; An of subsets in A we have that

Sn i=1 Ai^2 A.

We say that A is a -algebra if, additionally,

3í. for any countable collection fAigi 2 N of subsets in A, we have [ i 2 N Ai^ 2 A:

The family of all subsets of a set X is obviously a -algebra. Given C a family of subsets of X we deÖne the -algebra  (C) generated by C as the smallest -algebra containing C. That is, as the intersection of all the -algebras containing C. This is always well deÖned and is in general smaller than the -algebra of all subsets of X.

Exercise 6 Prove that the intersection of all the -algebras containing C is indeed a -algebra.

If X is a topological space, the -algebra generated by open sets is called the Borel -algebra and is denoted by B (X). Observe that a Cantor set C introduced in (2.1) is the complement of a countable union of open intervals and, therefore, belongs to Borel -algebra B ([0; 1]).

2.2 Measures and -algebras 9

1.  is -additive () For any increasing sequence fAngn 2 N  F (i.e., An  An+1) we have

n^ lim!1 ^ (An) =^ ^ (A)^ where^ A^ :=^

[

n 2 N An:

(a)  is -additive =) For any decreasing sequence fAngn 2 N  F (i.e., An+1  An) such that  (A 1 ) < 1 we have

n^ lim!1 ^ (An) =^ ^ (A)^ where^ A^ :=^

\

n 2 N An: (b) If for any decreasing sequence fAngn 2 N  F such that An & ;, we have limn!1  (An) = 0, then  is -additive.

DeÖning a countably additive function on -algebras is non-trivial. It is usually easier to deÖne countably additive functions on algebras because the class of sequences fAngn 2 N  A such that

S

n 2 N An^ 2 A^ is smaller than in^ - algebras. Observe that, unlike what happens in -algebras,

S

n 2 N An^ needs not belong to A if A is only an algebra. For example, the standard length is a countably additive function on the algebra generated by Önite unions of intervals. The fact that this extends to a countably additive function on the corresponding -algebra (and therefore, that we can measure Cantor sets) is guaranteed by the following fundamental result.

Theorem 10 (CarathÈodoryís Theorem, [3, Theorem 1.5.6]) Let e be a countably additive function deÖned on an algebra A of subsets. Then e can be extended in a unique way to a countably additive function  on the -algebra F =  (A).

Remark 11 The -additivity of e cannot be removed as the following counter-example shows. Let A be the algebra of sets A  N such that either A or NnA is Önite. For Önite A, let (A) = 0, and for A with a Önite complement let (A) = 1. Then  is an additive, but not countably additive set function. Proof. It is clear that A is indeed an algebra.  (A [ B) =  (A) +  (B) is obvious for disjoint sets A and B if A is Önite. Finally, A and B in A cannot be inÖnite simultaneously being disjoint. If  was countably additive, we would have

 (N) =

X^1

n=

 (fng) = 0;

which is clearly a contradiction.

2.2 Measures and -algebras 10

Nevertheless, deÖning measures on R is easier, as the next subsection summarises.

2.2.1 Measures on R

In this subsection, we are going to gather some deÖnitions and results that, roughly speaking, state that a measure on R is completely determined by the value of that measure on intervals of the form (a; b], a  b.

DeÖnition 12 A distribution function F : R! R is a right-continuous increasing function. That is,

x  y ) F (x)  F (y) and lim x!a+^

F (x) = F (a) :

Let now J := f(a; b] : a  b 2 Rg and let F : R! R be a distribution function. DeÖne  : J ! [0; 1 ] (a; b] 7 ! F (b) F (a):

Then, one can prove that  thus deÖned is a -additive and -Önite function on J. Moreover, there exists a unique -additive extension of  onto B (R). That is,

Theorem 13 A distribution function F : R! R determines a measure  on (R; B (R)) by means of the formula

 ((a; b]) = F (b) F (a); (a; b] 2 J:

A natural question now arises. Can we obtain any measure on (R; B (R)) from a distribution function? The answer is no, but almost any of them. Observe that the measure deÖned through (2.2) is Önite on any bounded interval. These are precisely the measures we can generate by means of distribution functions. They are called Lebesgue-Stieljes measures.

DeÖnition 14 A Lebesgue-Stieljes measure is a measure  on (R; B (R)) such that, for any bounded A 2 B (R),  (A) < 1.

Proposition 15 Let  be a Lebesgue-Stieljes measure. Then, there exists a distribution function F such that

8 a < b 2 R; F (b) F (a) =  ((a; b]) : (2.3)

2.2 Measures and -algebras 12

3. Absolutely continuous measures. Let f : R! R be a continuous function. For any subinterval I  R deÖne

 (I) :=

Z

I

f (y) dy:

Then  deÖnes a -Önitely additive function on the algebra of Önite unions of subintervals of R and thus extends uniquely to a measure on B(R). Indeed, a possible distribution function F associated to  is

F (x) =

Z (^) x

0

f (y) dy:

4. Normal law. The probability measure given by the distribution func- tion F (x) =

p 2 

Z (^) x

ey (^2) = 2 dy

is called the standard normal law and is denoted by N (0; 1).

5. Measures on spaces of sequences. Let + k denote the set of inÖnite sequences of k symbols. That is, and element a 2 + k is a sequence a = (a 0 ; a 1 ; :::) with ai 2 f 0 ; 1 ; :::; k 1 g. For any given Önite block (x 0 ; :::; xn 1 ) of length n with xi 2 f 0 ; 1 ; :::; k 1 g, let

Ix 0 :::xn 1 :=

a 2 + k : ai = xi; i = 0; :::; n 1

denote the set of all inÖnite sequences which start precisely with the prescribed Önite block (x 0 ; :::; xn 1 ). We call this a cylinder set of order n. Let

A = fÖnite unions of cylinder setsg :

Exercise 16 Show that A is an algebra of subsets of + k.

Fix now k numbers fp 0 ; :::; pk 1 g  [0; 1] such that p 0 + ::: + pk 1 = 1 and deÖne a function  : A! R+ on the algebra of cylinder sets by



Ix 0 :::xn 1

Yn 1 i=0 pxi^ :

Exercise 17 Prove that  is -additive.

Therefore, the function  extends uniquely to a measure on the - algebra F =  (A).

2.3 Integration 13

2.3 Integration

Integration with respect to a measure can be regarded as a powerful gen- eralization of the standard Riemann integral. In this section, we are going to review the basics of integration with respect to an arbitrary measure. Before, we need to introduce some deÖnitions. Let (X; F; ) be a measure space and let A 2 F. We deÖne the char- acteristic function (^1) A as

(^1) A (x) =

1 if x 2 A 0 if x = 2 A:

On the other hand, a simple or elementary function  : X! R is a function that can be written in the form

 =

X^ n

i=

ci (^1) Ai

for some constants ci 2 R and some disjoint measurable sets Ai 2 F, i = 1 ; :::; n. The integral of a simple function  with respect to the measure  is deÖned in a straightforward manner as Z d :=

X^ n

i=

ci (Ai) :

The idea is to extend this integral to more general functions. More con- cretely, we can deÖne the integral of a measurable function. Recall that a function f : X! R is measurable if f ^1 (I) 2 F for any I 2 B (R). If (X; F; ) is a probability space (i.e.,  is a probability), measurable functions are usually called random variables.

Exercise 18 Let f : X! R+ be a measurable function. Show that f is the (pointwise) limit of an increasing sequence of elementary functions. Hint: deÖne, for any n 2 N,

n :=

nX 2 n 1

k=

k 2 n^

(^1) f k 2 n^ f^ ^ k 2 +1n^ g^

• n (^1) fnf g:

The integral of a general, measurable, non-negative function f : R! R+ can be deÖned in two equivalent ways. On the one hand, Z f d := sup

Z

d :  is simple,   f

2.3 Integration 15

2. If A 2 F is such that  (A) = 0, then Z

A

f d :=

Z

(^1) Af d = 0:

That is, the integral of f over a set of measure 0 is 0. This is true even if f takes the values 1 on A. That is, if A contains singularities of f. Recall that we say that a point x = a is a singularity if f (a) = 1. For example, if a non-negative function f  0 is integrable,

R

f d < 1 , then we can say that  (fx : f (x) = 1g) = 0.

3. If f  0 and

R

f d = 0, then  (fx : f (x) > 0 g) = 0.

4. If f  g, then (^) Z f d 

Z

gd:

5. Z

f d 

Z

jf j d: (2.5)

Indeed,

R

f d =

R

f +d

R

f d and Z f d =

Z

f +d

Z

f d 

Z

f +d +

Z

f d

Z

f +d +

Z

f d =

Z

jf j d:

By (2.5), we can characterise L^1 (X; ) as the space of measurable functions f : X! R such that Z jf j d < 1 :

In general, for p  1 , we introduce the spaces Lp^ (X; ) as the space of measurable functions such that

R

X jf^ j

p (^) d < 1 , where two functions are identiÖed if they di§er, at most, on a set of zero measure. Lp^ (X; ) is a Banach space with the norm kf kp := (

R

X jf^ j

p (^) d) 1 =p.

Example 21 Let f : [0; 1]! R be given by

f (x) =

0 if x 2 Q 1 if x = 2 Q:

2.3 Integration 16

It is well known that this function is not Riemann integrable because the limit of the upper and lower Riemann sums do not coincide. However, as a function measurable with respect to the Lebesgue measure , f is simple: it values 0 on the measurable set Q and values 1 on the measurable set [0; 1]nQ. The set of rational numbers Q has zero Lebesgue measure because Q is countable. Therefore ([0; 1]nQ) = 1 and Z

[0;1]

f d =  ([0; 1]nQ) = 1:

3.1 Invariant measures: deÖnitions and examples 18

we deÖne T n^ (x) = T  : : :n)  T (x)

and T 0 = Id, the identity on X.

3.1.1 Examples

1. Dirac measures on Öxed points. If T : X! X is a measurable map and p a Öxed point of T , T (p) = p, then the Dirac measure p is invariant. Indeed, let A 2 F be a arbitrary measurable set. We have to prove that p

T ^1 (A)

= p (A) : (3.1) We consider two cases. First of all, suppose p 2 A so that p (A) =

1. In this case p 2 T ^1 (A) clearly so p

T ^1 (A)

= 1 and (3.1) holds. Secondly, suppose that p 2 = A. Then p (A) = 0 and we also have p 2 = T ^1 (A) because if p 2 T ^1 (A) then p = T (p) 2 T

T ^1 (A)

 A, which would be a contradiction. Therefore p 2 = T ^1 (A), p

T ^1 (A)

= 0, and (3.1) holds again.

2. Dirac measures on periodic orbits. Let T : X! X be a mea- surable map and let P = fa 1 ; :::; ang be a periodic orbit with minimal period n. That is, T (ai) = ai+1 for i = 1; :::; n 1 and T (an) = a 1. Let  1 ; :::; n be constants such that i 2 (0; 1) and

Pn i=1 i^ = 1. Consider the measure P (A) =

X

fi:ai 2 Ag

i:

Exercise 22 Show that P is invariant if and only if i = 1=n for every i = 1; :::; n.

3. Circle rotations.

Proposition 23 Let T : S^1! S^1 be a circle rotation, T (x) = x + for some 2 R. The Lebesgue measure is invariant.

Proof. T is just a translation and Lebesgue measure is invariant under translations. However, depending on the value of , there may be other invariant measures. If 2 = is rational then all points x 2 S^1 are periodic of the same period and, therefore, T admits also inÖnitely many distinct

3.2 PoincarÈís recurrence Theorem 19

Dirac measures on the periodic orbits (see Example 26). If 2 = is ir- rational, then all orbits are dense in S^1 (Example 26) and the Lebesgue measure is the unique invariant measure of T.

4. Measure-preserving áows in Rn. Let U  Rn^ be an open set and v : U! Rn^ a Cr^ vector Öeld, r  1. Consider the di§erential equation

x _ = v(x): (3.2)

Suppose that, for every p 2 U , there exists a (unique) solution x : R! U of (3.2) with initial condition p, which means that, x_t = v (xt) and xt=0 = p. For any t 2 R, we deÖne the map 't : U! U by 't (p) = xt where x : R! U is the solution of (3.2) with initial condition x 0 = p. Basic results of ordinary di§erential equations show that, for every t, the map 't is a Cr^ di§eomorphism and the family of maps 't : U! U deÖnes a one-parameter group, i.e., 't=0 = Id (the identity) and 't+s = 't  's for any t; s 2 R. Moreover, by Liouvilleís formula, det

@'t @xi^ (p) = exp

Z (^) t

0

div v ('s(p)) ds

for any p 2 U and t. Hence, if we assume div v = 0 , we have det @' @xti (p) = 1 and 't preserves the n-dimensional volume (or Lebesgue measure). Hamiltonian vector Öelds are examples of vector Öelds that satisfy div v = 0. Recall that a vector Öeld is called Hamiltonian if n = 2m is an even number and there exists a function H : U! R such that, denoting the points in Rn^ as (q 1 ; :::; qm; p 1 ; :::; pm),

v =

@H

@p 1

@H

@pm

@H

@q 1

@H

@qm

Exercise 24 Complete the proof and show that div v = 0 implies that the áow 't associated to v preserves the Lebesgue measure.

3.2 PoincarÈís recurrence Theorem

Invariant measures play a fundamental role in dynamics. As a Örst example, we state and prove the following famous result by PoincarÈ which implies that recurrence is a generic property of orbits of measure-preserving dynam- ical systems.