A comment on ergodic theorem for amenable groups

TL;DR

This paper extends the ergodic theorem for amenable groups by relaxing the temperedness condition on F{46}lner sequences, replacing it with a mixing condition on the function.

Contribution

It introduces a new ergodic theorem for amenable groups that does not require tempered F{46}lner sequences, relying instead on a mixing condition.

Findings

Ergodic theorem holds without tempered F{46}lner sequences.

A mixing condition on functions replaces the need for temperedness.

The result broadens applicability of ergodic theorems to more general group actions.

Abstract

We prove a version of ergodic theorem for an action of an amenable group, where a F{\o} lner sequence needs not to be tempered. Instead, it is assumed that a function satisfies certain mixing condition.

Equations

$${\underline{d}}_F(S)=\underset{g\in G}{\inf }\frac{|S\cap Fg|}{|F|}\text{and}{\overline{d}}_F(S)=\underset{g\in G}{\sup }\frac{|S\cap Fg|}{|F|}.$$

$$\underline{d}(S)$$

$$\overline{d}(S)$$

$$\underline{d}(S)=\underset{n\to \infty }{\lim }{\underline{d}}_{F_n}(S)\text{ and }\overline{d}(S)=\underset{n\to \infty }{\lim }{\overline{d}}_{F_n}(S).$$

$$\left|{\int }_{B}fd{\mu }_B-\int fd\mu \right|<\epsilon ,$$

$$\left|{\mu }_B(A)-\mu (A)\right|<\epsilon$$

$$|\mu (A\cap B)-\mu (A)\mu (B)|<\epsilon \mu (B).$$

$$\underset{n\to \infty }{\lim }\frac{1}{|F_n|}\sum _{g\in F_n}f(gx)=\int fd\mu \mu \text{-a.e.}$$

$$\mu \left(\left\{x:|M_n(x)|>\epsilon \right\}\right)⩽2\exp \left(\frac{-{\epsilon }^2}{2{\sum }_{k=1}^n{d}_k^2}\right)$$

$$Y_H=\sum _{g\in H}f\circ g{\overline{Y}}_H=\frac{1}{|H|}{Y}_H$$

$$Y_k^{(n,i)}=M_k^{(n,i)}+N_k^{(n,i)},$$

$$M_k^{(n,i)}=\sum _{j=1}^k(Y_j^{(n,i)}-E(Y_j^{(n,i)}|{\mathcal{F}}_{j-1}^{(n,i)}))$$

$$N_k^{(n,i)}$$

$$|M_k^{(n,i)}-M_{k-1}^{(n,i)}|$$

$${\overline{M}}^{(n,i)}=\frac{1}{|H^{(n,i)}|}{M}^{(n,i)},{\overline{N}}^{(n,i)}=\frac{1}{|H^{(n,i)}|}{N}^{(n,i)}$$

$$\begin{gathered} \mu \left(\left\{x:\left|{\overline{Y}}_{F_n}(x)-E{\overline{Y}}_{F_n}\right|>\epsilon \right\}\right) \\ ⩽\mu \left(\left\{x:\sum _{i=0}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{Y}}_{H^{(n,i)}}(x)-E{\overline{Y}}_{H^{(n,i)}}\right|>\epsilon \right\}\right) \\ ⩽\mu ({\mathcal{M}}_n)+\mu ({\mathcal{N}}_n), \end{gathered}$$

$${\mathcal{M}}_n=\left\{x:\begin{array}{rl}& \frac{\left|H^{(n,0)}\right|}{|F_n|}\left|{\overline{Y}}_{H^{(n,0)}}(x)-E{\overline{Y}}_{H^{(n,0)}}\right|+\\ & +\sum _{i=1}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{M}}^{(n,i)}(x)-E{\overline{M}}^{(n,i)}\right|>\epsilon /2\end{array}\right\}$$

$${\mathcal{N}}_n=\left\{x:\sum _{i=1}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{N}}^{(n,i)}(x)-E{\overline{N}}^{(n,i)}\right|>\epsilon /2\right\}$$

$$\left|E\left(f\circ g_{i_j}|{\mathcal{F}}_{j-1}^{(n,i)}\right)-\int fd\mu \right|<\epsilon /2$$

$$\left|\left({\overline{N}}^{(n,i)}(x)-E{\overline{N}}^{(n,i)}\right)\right|⩽\frac{1}{H^{(n,i)}}\sum _{j\in H^{(n,i)}}\left|E\left(f\circ g_{i_j}|{\mathcal{F}}_{j-1}^{(n,i)}\right)-\int fd\mu \right|<\epsilon /2$$

$$\sum _{i=1}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{N}}^{(n,i)}(x)-E{\overline{N}}^{(n,i)}\right|<\epsilon /2,$$

$$\frac{\left|H^{(n,0)}\right|}{|F_n|}<\epsilon /6{\Vert f\Vert }_{\infty }$$

$$\frac{\left|H^{(n,0)}\right|}{|F_n|}\left|{\overline{Y}}_{H^{(n,0)}}(x)-E{\overline{Y}}_{H^{(n,0)}}\right|⩽\frac{\left|H^{(n,0)}\right|}{|F_n|}\cdot 2{\Vert f\Vert }_{\infty }<\epsilon /3$$

$$\begin{gathered} \mu \left(\left\{x:\left|{\overline{M}}^{(n,i)}(x)-E{\overline{M}}^{(n,i)}\right|>\epsilon /6\right\}\right)= \\ =\mu \left(\left\{x:\left|M^{(n,i)}(x)-E{M}^{(n,i)}\right|>\epsilon /6\left|H^{(n,i)}\right|\right\}\right)⩽2{\gamma }^{|H^{(n,i)}|} \end{gathered}$$

$$X_{\epsilon }=\left\{x:\exists i=1,...,r\left|{\overline{M}}^{(n,i)}(x)-E{\overline{M}}^{(n,i)}\right|>\epsilon /6\right\}$$

$$\mu (X_{\epsilon })⩽2r\cdot {\gamma }^{{\min }_i|H^{(n,i)}|}.$$

$$\sum _{i=1}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{M}}^{(n,i)}(x)-E{\overline{M}}^{(n,i)}\right|⩽\epsilon /6,$$

$$\frac{\left|H^{(n,0)}\right|}{|F_n|}\left|{\overline{Y}}_{H^{(n,0)}}(x)-E{\overline{Y}}_{H^{(n,0)}}\right|+\sum _{i=1}^r\frac{\left|H^{(n,i)}\right|}{|F_n|}\left|{\overline{M}}^{(n,i)}(x)-E{\overline{M}}^{(n,i)}\right|>\epsilon /2$$

$$\mu ({\mathcal{M}}_n)⩽2r\cdot {\gamma }^{{\min }_i|H^{(n,i)}|}⩽2r{\left({\gamma }^{\beta }\right)}^{|F_n|}=:2r{\alpha }^{|F_n|}$$

$$\mu \left(\underset{n}{\mathrm{limsup}}{\mathcal{M}}_n\right)=0,$$

$$\begin{array}{r}\mu \left(\left\{x:\forall N\exists n⩾N\left|{\overline{Y}}_{F_n}(x)-\int fd\mu \right|>\epsilon \right\}\right)⩽\\ ⩽\mu \left(\underset{n}{\mathrm{limsup}}{\mathcal{M}}_n\right)+\mu \left(\underset{n}{\mathrm{limsup}}{\mathcal{N}}_n\right)=0,\end{array}$$

Full text

Research of both authors is supported from resources for science in years 2013-2018 as research project (NCN grant 2013/08/A/ST1/00275, Poland)

In [3] E.Lindenstrauss proved the ergodic theorem for actions of amenable groups, which is commonly used. The Følner sequence along which ergodic averages converge to an invariant function must satisfy the condition of being tempered. Since every Følner sequence has a subsequence which is tempered, the theorem is sufficient for many applications. In [2] it was shown that for the Bernoulli groups shift the assumption that the Følner sequence is tempered may be relaxed if one considers frequency of visits in a cylinder set. The aim of the current paper is to push further in this direction and ivestigate in what circumstances temperedness is not necessary.

Let $G=\{g_{1},g_{2},...\}$ be an amenable group and $(F_{n})_{n\in\mathbb{N}}$ a Følner sequence in $G$. Assume that for every $\alpha\in[0,1)$ the series $\sum_{n=1}^{\infty}\alpha^{|F_{n}|}$ converges. Since it is already satisfied if $|F_{n}|$ strictly increases, the assumption is much weaker than temperedness. For a finite nonempty set $F\subset G$ and any set $S\subset G$ we denote

[TABLE]

The lower and upper Banach densities of $S$ are defined by formulas:

[TABLE]

If $(F_{n})$ is a Følner sequence then we also have

[TABLE]

The following lemma was proved in [2].

1. (1)

$\overline{d}_{(F_{n})}(D_{0})\leqslant\delta$, 2. (2)

$\underline{d}_{(F_{n})}(D_{i})>0$* for every $i=1,...,r$,* 3. (3)

for every $i=1,...,r$, if $g,h\in D_{i}$ then $Kg\cap Kh=\varnothing$.

Let $G$ act via measure preserving transformations on a probability space $(X,\mu)$. To simplify the notation we will indentify $G$ with the related group of automorphisms and write $gx$ for the outcome of the action of an automorphism associated to $g$ on $x$, and $f\circ g$ for the composition of a function $f$ and the automorphism. According to [1], for any action of $\mathbb{Z}$ one can find a Følner sequence (with cardinalities of sets increasing slowly) such that the ergodic averages with respect to that sequence fail to converge for some function. Therefore, some constraints must be put on the function, whose ergodic averages we study. For $\mathcal{F}\subset L^{1}(\mu)$ let $\sigma(\mathcal{F})$ denote the smallest sub-$\sigma$-algebra with respect to which every $f\in\mathcal{F}$ is measurable.

[TABLE]

where $\mu_{B}$ is the conditional measure on $B$.

A set $A$ is $\varepsilon$-independent from $\Sigma_{0}$ if its charachteristic function $\mathbbm{1}_{A}$ is, i.e., for all $B\in\Sigma_{0}$ such that $\mu(B)>0$,

[TABLE]

or, in other words,

[TABLE]

[TABLE]

Note that the neutral element of $G$ belongs to $K$ for non-constant $f$.

Before the proof let us recall the following concentration inequality.

[TABLE]

[TABLE]

Then $\int fd\mu=E\overline{Y}_{H}$.

Let $H^{(n,i)}_{k}=\{g_{i_{1}},...,g_{i_{k}}\}$ be a set of $k$ first elements of $G$ belonging to $H^{(n,i)}=F_{n}\cap D_{i}$. Let $\mathcal{F}^{(n,i)}_{k}=\sigma(\{f\circ g:g\in H^{(n,i)}_{k}\})$. Clearly, $\{\mathcal{F}^{(n,i)}_{k}\}$ is a finite filtration and for each pair $(n,i)$ the process $Y^{(n,i)}_{k}=Y_{H^{(n,i)}_{k}}$ is adapted to it. Let $\mathcal{F}_{0}^{(n,i)}$ be the trivial $\sigma$-algebra in $X$ and let $Y_{0}^{(n,i)}=0$ a.s. By Doob’s decomposition,

[TABLE]

where

[TABLE]

is a martingale and

[TABLE]

is a predictable process, i.e. each $N_{k}^{(n,i)}$ is $\mathcal{F}_{k-1}^{(n,i)}$-measurable. Then $EM_{k}^{(n,i)}=0$ and $EN_{k}^{(n,i)}=k\int fd\mu$. Note also, that

[TABLE]

Similarly, we write $Y_{H^{(n,i)}}=M^{(n,i)}+N^{(n,i)}$ and we denote:

[TABLE]

We will estimate the following quantity:

[TABLE]

where

[TABLE]

and

[TABLE]

Below we estimate the second summand. For each $j$,

[TABLE]

almost surely, because $f\circ g_{i_{j}}$ is $\varepsilon/2$-independent from $\mathcal{F}_{j-1}^{(n,i)}$. Then

[TABLE]

almost surely. Hence also the average satisfies

[TABLE]

so $\mu(\mathcal{N}_{n})=0$.

Now we will estimate $\mu(\mathcal{M}_{n})$. We have $\overline{d}_{(F_{n})}(D_{0})\leqslant\varepsilon/7\left\|f\right\|_{\infty}$, so for large $n$,

[TABLE]

Clearly, $Y_{H^{(n,0)}}$ is a sum of functions with bounded range, hence

[TABLE]

for large $n$.

By Azuma-Hoeffding inequality, for each $i=1,...,r$ it holds that

[TABLE]

for $\gamma=\exp\left(-\frac{\varepsilon^{2}}{8||f||_{\infty}^{2}}\right)<1$ (recall that $|M_{k}^{(n,i)}-M_{k-1}^{(n,i)}|$ is bounded almost surely). Denoting

[TABLE]

we obtain

[TABLE]

Thus, outside $X_{\varepsilon}$ not only $\left|\overline{M}^{(n,i)}(x)-E\overline{M}^{(n,i)}\right|\leqslant\varepsilon/6$, but also the weighted average satisfies the same inequality

[TABLE]

so it is only the set $X_{\varepsilon}$ on which it may happen that

[TABLE]

By positive density of each $D_{i}$, there is a positive number $\beta$ such that $\min_{i}|H^{(n,i)}|>\beta|F_{n}|$ for large $n$. We obtain

[TABLE]

By the assumption on the Følner sequence $(F_{n})$, the series $\sum_{n}\alpha^{|F_{n}|}$ converges. By Borel-Cantelli lemma we obtain that

[TABLE]

so

[TABLE]

Finally, after taking an appropriate countable intersection we obtain

[TABLE]

∎