Posted by: yglima | March 29, 2012

CF3: infinite ergodic theory

This post intends to treat the classical results on infinite ergodic theory, specifically Hopf’s ratio ergodic theorem, the non-existence of Birkhoff’s theorem, Aaronson’s theorem, rational ergodicity and law of large numbers. We won’t provide proofs for these results and for that we refer the reader to the book An Introduction to Infinite Ergodic Theory of Jon Aaronson.

Let {(X,\mathcal A,\mu,F)} be a measure-preserving system: {(X,\mathcal A,\mu)} is a measure space, {\mu} a {\sigma}-finite measure and {F} is a measurable transformation on {X} that is invariant under {\mu}. Whenever {\mu(X)<\infty}, we have Poincaré’s recurrence theorem. If {\mu(X)=\infty}, this is not true in general and one has to assume an additional condition, described by the

Definition 1 We say {\mu} is conservative if {\mu(A)=0} whenever {A\in\mathcal A} is such that {\{F^{-n}A\}_{n\ge 0}} are pairwise disjoint.

Exercise 1 Prove that a conservative measure-preserving system satisfies Poincaré’s recurrence theorem.

Conservativity is also a necessary condition: the translation {x\mapsto x+1}, {x\in\mathbb Z}, is invariant for the counting measure on {\mathbb Z} but the orbit of every point scapes to infinity. This is indeed, if one assumes ergodicity and invertibility, the only example.

Lemma 2 An invertible ergodic measure-preserving system is non-conservative if and only if it is isomorphic to the translation {x\mapsto x+1}, {x\in\mathbb Z}, with the counting measure on {\mathbb Z}.

Proof: Assume {(X,\mathcal A,\mu,F)} is an invertible ergodic non-conservative measure-preserving system and let {A\in\mathcal A} such that {\{F^{-n}A\}_{n\ge 0}} are pairwise disjoint. Because {F} is invertible, {\{F^{-n}A\}_{n\in\mathbb Z}} are pairwise disjoint and thus, by ergodicity,

\displaystyle X=\bigcup_{n\in\mathbb Z}F^{-n}A.

Again by ergodicity, {A} is a {\mu}-atom (non trivial subsets of {B} of {A} give rise to non trivial {F}-invariant sets {\bigcup_{n\in\mathbb Z}F^{-n}B}). This guarantees the map {n\mapsto F^nA} is an isomorphism from {\mathbb Z} to {X}. \Box

In the non-invertible situation, there are many other examples. See the end of this paper of J. Aaronson and T. Meyerovitch.

From now on, we always assume {(X,\mathcal A,\mu,F)} is ergodic and conservative. Let {\phi:X\rightarrow {\mathbb R}} be a measurable function. A successful area in ergodic theory deals with the convergence of the averages {n^{-1}\cdot\sum_{k=0}^{n-1}\phi\left(F^kx\right)}, {x\in X}, when {n} goes to infinity. The well known Birkhoff’s theorem states that, if {\mu(X)<\infty}, such limit exists for almost every {x\in X} whenever {\phi} is a {L^1}-function. This is not the case when {\mu} is infinite. Indeed, if {\mu(X)=\infty}, these averages converge to zero for almost every {x\in X}. We give an idea why this is so: if {(X,\mathcal A,\mu,F)} is a measure-preserving system with {\mu(X)<\infty}, Birkhoff’s theorem asserts that

\displaystyle \dfrac{1}{n}\sum_{k=0}^{n-1}\phi\left(F^kx\right)\longrightarrow\dfrac{1}{\mu(X)}\int_X\phi d\mu\ \ \text { as }n\rightarrow\infty

for {\mu}-almost every {x\in X}. Heuristically, if we take {\mu(X)=\infty} in the right hand side of the above convergence, then the Birkhoff averages converge to zero. This argument demands a little care, and indeed is more convenient to conclude it as a corollary of Hopf’s ratio ergodic theorem (see below). At this point, it is natural to ask if there exists some “appropriate” rate of convergence: is there a normalizing sequence of constants {(a_n)} such that {{a_n}^{-1}\cdot\sum_{k=0}^{n-1}\phi\left(F^kx\right)} converges almost surely? And the answer is no!

Theorem 3 (Aaronson) Let {(X,\mathcal A,\mu,F)} be a measure-preserving system with {\mu(X)=\infty}, and let {(a_n)} be a sequence of positive real numbers. Then, for every {\phi\in L^1(X,\mathcal A,\mu)} such that {\phi\ge 0} and {\int_X\phi d\mu>0},

\displaystyle \limsup_{n\rightarrow\infty}\dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{a_n}=\infty\ \text{ a.e }\ \ \text{ or }\ \ \liminf_{n\rightarrow\infty}\dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{a_n}=0\ \text{ a.e}.

This means that any attempt of normalization will under or overestimate the behavior of Birkhoff sums. Nevertheless, the Birkhoff sums fluctuate in the same proportional rate, according to the following result.

Theorem 4 (Hopf’s ratio ergodic theorem) Let {(X,\mathcal A,\mu,F)} be a measure-preserving system. Then, for every {\phi,\psi\in L^1(X,\mathcal A,\mu)} such that {\psi\ge 0} and {\int_X\psi d\mu>0},

\displaystyle \dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{\sum_{k=0}^{n-1}\psi\left(F^kx\right)}\longrightarrow \dfrac{\int_X\phi\,d\mu}{\int_X\psi d\mu}\ \ \text { as }n\rightarrow\infty

for {\mu}-almost every {x\in X}.

Exercise 2 Prove, using Hopf’s ratio ergodic theorem, that if {(X,\mathcal A,\mu,F)} is a measure-preserving system with {\mu(X)=\infty} then, for every {\phi\in L^1(X,\mathcal A,\mu)},

\displaystyle \dfrac{1}{n}\sum_{k=0}^{n-1}\phi\left(F^kx\right)\longrightarrow 0\ \ \text { as }n\rightarrow\infty

for {\mu}-almost every {x\in X}.

Hopf’s ratio ergodic theorem is an indication that some sort of regularity might exist and it might still be possible, for a specific sequence {(a_n)}, that the averages oscillate without converging to zero or infinity and so one can hope for a summability method that smooths out the fluctuations and forces convergence.

1. Law of large numbers

More generally, one can hope for a law of large numbers.

Definition 5 A law of large numbersfor a measure-preserv\-ing system {(X,\mathcal A,\mu,F)} is a function {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]} such that, for any {A\in\mathcal A}, the equality

\displaystyle L(\chi_A(x),\chi_A(Fx),\chi_A(F^2x),\ldots)=\mu(A)

holds for {\mu}-almost every {x\in X}.

One can see the function {L} as a sort of blackbox: given the input of hittings of a generic point {x\in X} to a fixed set {A\in\mathcal A}, the output is the measure of {A}. For example, if {\mu(X)=1}, the function {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]} defined by

\displaystyle L(x_0,x_1,\ldots)=\left\{ \begin{array}{rl} \displaystyle\lim_{n\rightarrow\infty}\dfrac{1}{n}\sum_{k=0}^{n-1}x_k\ ,&\text{ if the limit exists,}\\ &\\ 0\ ,&\text{ otherwise} \end{array} \right.

is a law of large numbers. The infinite measure situation is quite different: there are systems with no law of large numbers. For example, let {F} be squashable: there is {G:(X,\mathcal A)\rightarrow(X,\mathcal A)}, commuting with {F}, such that

\displaystyle \mu(G^{-1}A)=c\cdot\mu(A)\ \ \text{for all }A\in\mathcal A, \ \ \ \ \ (1)

for some {c\not=1}. If {F} had a law of large numbers, say {L}, then for {\mu}-almost every {x\in X} we would have

\displaystyle \begin{array}{rcl} \mu(A)&=&L(\chi_A(Gx),\chi_A(FGx),\ldots)\\ &&\\ &=&L(\chi_A(Gx),\chi_A(GFx),\ldots)\\ &&\\ &=&L(\chi_{G^{-1}A}(x),\chi_{G^{-1}A}(Fx),\ldots)\\ &&\\ &=&\mu(G^{-1}A), \end{array}

contradicting the assumption of squashability. See An Introduction to Infinite Ergodic Theoryfor more on squashable systems.

There are, fortunately, some conditions that guarantee the existence of law of large numbers.

2. Rational ergodicity

Given {A\in\mathcal A}, let {S_n(A):X\rightarrow{\mathbb N}} be the Birkhoff sum of the characteristic function {\chi_A} with respect to {F}.

Definition 6 A measure-preserving system {(X,\mathcal A,\mu,F)} is called rationally ergodic if there is a sweep-out set {A\in\mathcal A} with {0<\mu(A)<\infty} satisfying a Renyi inequality: there is {M>0} such that

\displaystyle \int_A S_n(A)^2d\mu\ \le \ M\cdot\left(\int_A S_n(A)d\mu\right)^2

for every {n\ge 1}. We say {(X,\mathcal A,\mu,F)} is rationally ergodic along a subsequence of iterates if the above inequality holds for a subsequence {(n_k)} of positive integers.

By sweep-out we mean that every point of {X} eventually enters in {A}, that is

\displaystyle X=\bigcup_{n\ge 0}F^{-n}A.

Definition 7 {(X,\mathcal A,\mu,F)} is called weakly homogeneous if there is a sequence {(a_{n_k})} of positive real numbers such that, for all {\phi\in L^1(X,\mathcal A,\mu)},

\displaystyle \dfrac{1}{N}\sum_{k=1}^N\dfrac{1}{a_{n_k}}\sum_{j=0}^{n_k-1}\phi\left(F^jx\right)\ \longrightarrow\ \int_X\phi d\mu \ \ \text { as }n\rightarrow\infty \ \ \ \ \ (2)

for {\mu}-almost every {x\in X}.

{(a_{n_k})} is called a return sequence of {F} and it is unique up to asymptotic equality.

Theorem 8 (Aaronson) Every measure-preserving system {(X,\mathcal A,\mu,F)} that is rationally ergodic along a subsequence of iterates is weakly homogeneous. More specifically, every subsequence {(a_{n_k})} can be refined to a further subsequence such that (2) holds for {\mu}-almost every {x\in X}.

Theorem 8 also gives the explicit description of the normalizing constants:

\displaystyle a_{n_k}=\dfrac{1}{\mu(A)^2}\int_A S_{n_k}(A)d\mu =\dfrac{1}{\mu(A)^2}\sum_{j=0}^{n_k-1}\mu\left(A\cap F^{-j}A\right). \ \ \ \ \ (3)

Observe that weak homogeneity defines an explicit law of large numbers {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]} by

\displaystyle L(x_0,x_1,\ldots)=\left\{ \begin{array}{rl} \displaystyle\lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{k=1}^N\dfrac{1}{a_{n_k}}\sum_{j=0}^{n_k-1}x_j\ ,&\text{ if the limit exists,}\\ &\\ 0\ ,&\text{ otherwise.} \end{array} \right.

The next post will focus on a result in collaboration with P. Cirilo and E. Pujals in which we construct a class of cylinder flows that are rationally ergodic along a subsequence of iterates (and thus have explicit law of large numbers).

Previous posts: CF0CF1, CF2.


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