Posted by: yglima | December 14, 2009

## ERT4: Multiple Ergodic Averages

In this post we discuss, without proofs, convergence of multiple ergodic averages to give the reader a broader notion of the flavour of the results. The last two posts showed that recurrence is a natural phenomenon and occurs in a regular way. The next question is to ask for multiple recurrence: given a mps ${(X,\mathcal B,\mu,T)}$, ${A\in\mathcal B}$ such that ${\mu(A)>0}$ and a positive integer ${k}$, there exist positive integers ${a_1,\ldots,a_k}$ such that

$\displaystyle \mu(A\cap T^{-a_1}A\cap\cdots\cap T^{-a_k}A)>0\ \ \text{?}$

Formulated as it is, this question follows simply by multiple applications of Poincaré’s Recurrence Theorem (see ERT1): there exists ${n_1>0}$ such that ${\mu(A\cap T^{-n_1}A)>0}$. Letting ${A_1=A\cap T^{-n_1}A}$, there exists ${n_2>0}$ such that ${\mu(A_1\cap T^{-n_1}A_1)>0}$, which is the same as

$\displaystyle \mu\left(A\cap T^{-n_1}A\cap T^{-n_2}A\cap T^{-(n_1+n_2)}A\right)>0.$

Repeating the argument ${k}$ times, we obtain positive integers ${n_1,\ldots,n_k}$ such that

$\displaystyle \mu\left(\bigcap_{E\subset\{n_1,\ldots,n_k\}}T^{-S(E)}A\right)>0,$

where ${S(E)=\sum_{n\in E}n}$. This is much more than we wanted. In fact, applying the argument infinitely many times, we construct a sequence ${(n_k)_{k\ge 1}}$ of positive integers such that

$\displaystyle \mu\left(\bigcap_{E\in\mathcal F}T^{-S(E)}A\right)>0$

for every finite family ${\mathcal F}$ of subsets of ${\{n_1,n_2,\ldots\}}$. Unfortunately, we have no control in the ${n_k}$‘s, so that combinatorial applications are harder. It would be interesting if we had some regularity in them. For example, can they form an arithmetic progression? The answer is YES and this constitutes one of the pilars of Ergodic Ramsey Theory.

Theorem 1 (Furstenberg) If ${(X,\mathcal B,\mu,T)}$ is a mps, ${A\in\mathcal B}$ such that ${\mu(A)>0}$ and ${k}$ is a positive integer, then there exists a positive integer ${n}$ such that $\displaystyle \mu\left(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A\right)>0. \ \ \ \ \ (1)$

Obviously, the existence of such ${n}$ is equivalent to the existence of ${N>0}$ such that $\displaystyle \dfrac{1}{N}\sum_{n=1}^{N}\mu\left(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A\right)>0. \ \ \ \ \ (2)$

Taking ${f=\chi_A}$, the characteristic function of $A$,

$\displaystyle \mu\left(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A\right)=\int_X f\cdot T^nf\cdots T^{kn}fd\mu,$

where $Tf=f\circ T$, so that (2) is equivalent to

$\displaystyle \int_X\left(\dfrac{1}{N}\sum_{n=1}^{N}f\cdot T^nf\cdots T^{kn}f\right)d\mu>0.$

This inquires the analysis of the averages $\displaystyle f_N=\dfrac{1}{N}\sum_{n=1}^{N}f\cdot T^nf\cdots T^{kn}f,\ \ N>0. \ \ \ \ \ (3)$

Due to its nonsymmetry, instead of (3) we consider ${k}$ commuting transformations ${T_1,\ldots,T_k}$, all of them preserving ${\mu}$, and the averages $\displaystyle f_N=\dfrac{1}{N}\sum_{n=1}^{N}f\cdot{T_1}^nf\cdots {T_k}^nf, \ \ \ \ \ (4)$

from now on called multiple ergodic averages. Clearly, (3) is a special case of (4) considering ${T_i=T^i}$, ${i=1,2,\ldots,k}$. Although the purpose is the full generality of (4), it is natural first to investigate (3). Four situations deserve attention:

• If ${0\le f\le 1}$ and ${\int fd\mu>0}$, does (4) have positive ${\liminf}$?
• ${L^2}$-norm convergence.
• Pointwise convergence.
• What about convergence of multiple polynomial ergodic averages?

The first one was solved affirmatively by H. Furstenberg in the 1977 seminal paper Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.

Theorem 2 (Furstenberg) Let ${(X,\mathcal B,\mu,T)}$ be a mps and ${f\in L^\infty}$ be non-negative and satisfy ${\int fd\mu>0}$. Then, for any ${k\ge 1}$,
$\displaystyle \liminf_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^{N}\int f\cdot T^nf\cdots T^{kn}fd\mu>0.$

As we’ll see in forecoming posts, this actually proves Szemerédi’s Theorem, via a correspondence principle between sets of integers of positive density and measure-preserving systems. One year after, motivated by a topological analogue due to B. Weiss, Furstenberg and Y. Katznelson established in An ergodic Szemerédi theorem for commuting transformations an extension to the commutative case.

Theorem 3 (Furstenberg and Katznelson) Let ${(X,\mathcal B,\mu)}$ be a probability measure space and ${T_1,\ldots,T_k}$ commuting transformations, all of them preserving ${\mu}$. Then, for any non-negative ${f\in L^\infty}$ such that ${\int fd\mu>0}$,$\displaystyle \liminf_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^{N}\int f\cdot {T_1}^nf\cdots {T_k}^nfd\mu>0.$

This result, in addition to extending Furstenberg’s Theorem, implies a purely combinatorial multidimensional version of Szemerédi’s Theorem.

Theorem 4 (Multidimensional Szemerédi’s Theorem) Let ${E\subset{\mathbb Z}^d}$ be a subset with positive upper-Banach density and ${F\subset{\mathbb Z}^d}$ be any finite configuration. Then there are an integer ${r}$ and a vector ${u\in{\mathbb Z}^d}$ such that ${u+rF\subset E}$.

An interesting feature is that, until 2007, when hypergraph versions of Szemerédi’s Regularity Lemma were developed by T. Gowers, there was no combinatorial proof of this result.

After establishing positivity, we discuss ${L^2}$-norm convergence, solved in Nonconventional ergodic averages and nilmanifolds by B. Host and B. Kra.

Theorem 5 (Host and Kra) Let ${(X,\mathcal B,\mu,T)}$ be a mps and ${f_0,\ldots,f_k}$ be ${k+1}$ bounded measurable functions on ${X}$. Then

$\displaystyle \lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{n=1}^N f_0\cdot T^nf_1\cdots T^{kn}f_k$

exists in ${L^2}$.

Observe that we no longer have only one function, but $k+1$. Three years later, in 2008, T. Tao extended it to the commuting setup in the work Norm convergence of multiple ergodic averages for commuting transformations.

Theorem 6 (Tao) Let ${(X,\mathcal B,\mu)}$ be a probability measure space, ${T_1,\ldots,T_k}$ measure-preserving commuting transformations and ${f_0,\ldots,f_k}$ be ${k+1}$ bounded measurable functions on ${X}$. Then$\displaystyle \lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{n=1}^N f_0\cdot{T_1}^nf_1\cdots{T_k}^nf_k$

exists in ${L^2}$.

It is worth mentioning that this year T. Austin gave a new proof of it using classical ergodic theory (On the norm convergence of  nonconventional ergodic averages). A few is known about pointwise convergence, only that

$\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^{N}T^{an}f\cdot T^{bn}g$

converges almost surely, for any ${a,b\in{\mathbb Z}}$ and ${f,g\in L^\infty}$. This was obtained by J. Bourgain in Double recurrence and almost sure convergence.

Now consider polynomials ${p_1,\ldots,p_k}$ with integers coefficients and the limits

$\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N f_0\cdot{T_1}^{p_1(n)}f_1\cdots{T_k}^{p_k(n)}f_k.$

What is known? In terms of combinatorial appications, does it at least have positive ${\liminf}$ whenever ${f_0=\cdots=f_k=f}$ is a non-negative bounded function such that ${\int fd\mu>0}$? Yes… due to V. Bergelson and A. Leibman in the work Polynomial extensions of van der Waerden’s and Szemerédi’s theorems.

Theorem 7 (Bergelson and Leibman) Let ${(X,\mathcal B,\mu)}$ be a probability measure space, ${T_1,\ldots,T_k}$ measure-preserving commuting transformations, ${p_1,\ldots,p_k}$ polynomials with integer coefficients and ${f\ge 0}$ a bounded measurable functions on ${X}$ such that ${\int fd\mu>0}$. Then$\displaystyle \liminf_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\int f\cdot{T_1}^{p_1(n)}f\cdots{T_k}^{p_k(n)}fd\mu>0.$

In fact, they proved a more general result.

Theorem 8 (Bergelson and Leibman) Let ${(X,\mathcal B,\mu)}$ be a probability measure space, ${T_1,\ldots,T_k}$ measure-preserving commuting transformations, ${p_{11},\ldots,p_{1t}}$, ${p_{21},\ldots,p_{2t}}$,${\ldots}$, ${p_{k1},\ldots,p_{kt}}$ polynomials with integer coefficients and ${f\ge 0}$ a bounded measurable functions on ${X}$ such that ${\int fd\mu>0}$. Then$\displaystyle \liminf_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\int\left( f\cdot\prod_{1\le j\le t}{T_j}^{p_{1j}(n)}f\cdots \prod_{1\le j\le t}{T_j}^{p_{kj}(n)}f\right)d\mu>0.$

I could not find any result about ${L^2}$-norm convergence. Two works, Convergence of polyomial ergodic averages by Host and Kra and Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold by Leibman, both in 2005, proved the situation of multiple polynomial ergodic averages along one transformation.

Theorem 9 (Host, Kra and Leibman) Let ${(X,\mathcal B,\mu,T)}$ be a mps, ${p_1,\ldots,p_k}$ polynomials with integer coefficients and ${f_1,\ldots,f_k}$ bounded measurable functions on ${X}$. Then

$\displaystyle \lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{n=1}^N {T}^{p_1(n)}f_1\cdots{T}^{p_k(n)}f_k$

exists in ${L^2}$.

Last News: a few hours ago this paper was posted in arXiv by Q. Chu, N. Frantzikinakis and B. Host stating many cases of the $L^2$-norm convergence of multiple ergodic polynomial averages for commuting transformations.

Theorem 10 (Chu, Frantzikinakis and Host) Let ${(X,\mathcal B,\mu)}$ be a probability measure space, ${T_1,\ldots,T_k}$ measure-preserving invertible commuting transformations, ${p_1,\ldots,p_k}\in\mathbb Z[x]$ polynomials with different degrees and ${f_1,\ldots,f_k}$ bounded measurable functions on ${X}$. Then

$\displaystyle \lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{n=1}^N {T_1}^{p_1(n)}f_1\cdots{T_k}^{p_k(n)}f_k$

exists in ${L^2}$.

As every fresh result, it first needs to be checked in full details.

Previous posts: ERT0, ERT1, ERT2, ERT3.