[Paper Review] Pointwise recurrence for commuting measure preserving transformations

Idris Assani|arXiv (Cornell University)|Dec 18, 2013

Limits and Structures in Graph Theory12 references5 citations

TL;DR

This paper establishes the almost everywhere pointwise convergence of nonconventional ergodic averages for commuting measure-preserving transformations on a probability space. Using a novel approach based on dynamical systems models and invariant measures, it proves that averages of the form $\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{H}f_i(T_i^n x)$ converge a.e. for bounded functions $f_i$, solving a long-standing open problem in ergodic theory and providing a new proof of Bourgain's double recurrence theorem.

ABSTRACT

Let $(X,\mathcal{A}, μ)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$ converge a.e. for every function $f_i \in L^{\infty}(μ)$ .\\ As a consequence if $T_i = T^i$ for $1\leq i \leq H$ where $T$ is an invertible measure preserving transformation on $(X, \mathcal{A}, μ)$ then the averages $$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For $H=2$ it provides another proof of J. Bourgain's a.e. double recurrence theorem.

Motivation & Objective

• To establish the almost everywhere pointwise convergence of nonconventional ergodic averages involving multiple commuting measure-preserving transformations.
• To resolve a long-standing open question regarding the pointwise convergence of averages of the form $\frac{1}{N}\sum_{n=1}^{N}f_1(T^{n}x)\cdots f_H(T^{Hn}x)$.
• To provide a new proof of J. Bourgain's a.e. double recurrence theorem using a different method than characteristic factors or uniform Wiener-Wintner estimates.
• To extend the convergence result from free actions to general commuting measure-preserving systems via decomposition techniques.

Proposed method

• Utilizes a model dynamical system via B. Weiss's isomorphism theorem to represent the original measure-preserving system as a minimal, free $\mathbb{Z}^H$-action on a compact metrizable space.
• Introduces a probability measure $\nu$ on $X^H$ defined via a weighted sum of pushforwards of the diagonal measure $\mu_\Delta$ under iterates of the transformation $\Phi(z) = (T_1 z_1, \dots, T_H z_H)$.
• Applies the maximal ergodic theorem and maximal inequality to show $\nu$-a.e. convergence of the averages $M_N(F)(z) = \frac{1}{N}\sum_{k=1}^N F(\Phi^k z)$ for $F = \otimes f_i$.
• Transfers the $\nu$-a.e. convergence result back to the original measure space using the isomorphism, leveraging that $\nu$-null sets are $\mu_\Delta$-null.
• Uses decomposition of the space into periodic and aperiodic sets to extend convergence from free actions to general commuting systems.
• Applies the result to the case where $T_i = T^i$ for a single transformation $T$, recovering the convergence of $\frac{1}{N}\sum_{n=1}^N f_1(T^n x)\cdots f_H(T^{Hn}x)$ a.e.

Experimental results

Research questions

• RQ1 Do nonconventional ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^n x)\cdots f_H(T_H^n x)$ converge almost everywhere for commuting measure-preserving transformations?
• RQ2 Can the pointwise convergence of averages involving higher-order polynomials in the exponents be established using this method?
• RQ3 Does the result extend to non-free actions or systems with periodic components?
• RQ4 Can this approach be adapted to prove convergence of ergodic Hilbert transforms involving such averages?

Key findings

• The averages $\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^H f_i(T_i^n x)$ converge almost everywhere for all $f_i \in L^\infty(\mu)$ when the $T_i$ are commuting, invertible, measure-preserving transformations.
• The result holds even when the action generated by the $T_i$ is not free, by decomposing the space into periodic and aperiodic parts and applying the convergence result on each component.
• For the special case $T_i = T^i$, the convergence of $\frac{1}{N}\sum_{n=1}^N f_1(T^n x)\cdots f_H(T^{Hn}x)$ a.e. is established, solving a long-open problem in ergodic theory.
• The method provides an alternative proof of J. Bourgain’s a.e. double recurrence theorem without relying on characteristic factors or uniform Wiener-Wintner estimates.
• The convergence of the averages is shown first with respect to a measure $\nu$ on $X^H$, and then transferred to the diagonal measure $\mu_\Delta$ via measure-theoretic isomorphism.
• The result extends to systems where the transformations generate a nilpotent group, as shown in related work by M. Walsh and T. Austin, though the current method does not directly apply to polynomial sequences like $n^2$.