[Paper Review] Pointwise multiple averages for systems with two commuting transformations

Sebastián Donoso, Wenbo Sun|arXiv (Cornell University)|Sep 30, 2015

Limits and Structures in Graph Theory12 references4 citations

TL;DR

This paper establishes pointwise multiple ergodic average convergence for systems with two commuting transformations, using topological models and structural decomposition. It proves that for ergodic systems with commuting transformations S and T, the average $\frac{1}{N^3}\sum_{i,j,k=0}^{N-1} f_0(S^jT^k x)f_1(S^{i+j}T^k x)f_2(S^jT^{i+k}x)$ converges a.e. for bounded functions, and in the distal case, the simpler average $\frac{1}{N}\sum_{i=0}^{N-1} f_1(S^i x)f_2(T^i x)$ also converges a.e.

ABSTRACT

We show that if $(X,\mathcal{X},μ,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\] converges for $μ$-a.e. $x\in X$ as $N o \infty$ for $f_0,f_1, f_2\in L^\infty(μ)$. We also show that if $(X,\mathcal{X},μ,S,T)$ is a measurable distal system, the average \[ \frac{1}{N}\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \] converges for $μ$-a.e. $x\in X$ as $N o \infty$ for $f_1,f_2\in L^{\infty}(μ)$.

Motivation & Objective

• To establish pointwise convergence of multiple ergodic averages in systems with two commuting measure-preserving transformations.
• To extend the use of topological models—previously applied in single-transformation settings—to systems with two commuting transformations.
• To prove convergence results in the distal case, where additional structural properties simplify the analysis.
• To develop a new topological structure $N_{S,T}(X)$ that enables the construction of strictly ergodic models for such systems.
• To generalize previous pointwise convergence results, particularly those of Huang, Shao, and Ye, to the two-commuting-transformations setting.

Proposed method

• Construct a topological model $N_{S,T}(X)$ for any ergodic system with two commuting transformations $S$ and $T$, ensuring strict ergodicity.
• Use the existence of a strictly ergodic model to reduce the pointwise convergence problem to a topological setting where continuous functions are dense.
• Apply a factorization technique via intermediate factors $\mathcal{Z}_{S,R}(X)$ and $\mathcal{Z}_{T,R}(X)$, leveraging the structure of invariant $\sigma$-algebras.
• Use induction on the ordinal index $\theta$ in a transfinite sequence of isometric extensions to propagate the convergence property from base cases.
• Leverage the Birkhoff Ergodic Theorem on the common factor $X_R$ (associated with $R$) to handle the coupled dynamics of $S$ and $T$.
• Use a density argument in $L^1$ and conditional expectation estimates to control approximation errors and show convergence a.e.

Experimental results

Research questions

• RQ1 Does the triple average $\frac{1}{N^3}\sum_{i,j,k=0}^{N-1} f_0(S^jT^k x)f_1(S^{i+j}T^k x)f_2(S^jT^{i+k}x)$ converge pointwise a.e. for all bounded functions in an ergodic system with two commuting transformations?
• RQ2 Can the pointwise convergence of $\frac{1}{N}\sum_{i=0}^{N-1} f_1(S^i x)f_2(T^i x)$ be established in the distal case using topological models?
• RQ3 How can topological models be extended to handle multiple commuting transformations and their joint dynamics?
• RQ4 What structural properties (e.g., isometric extensions, magic extensions) are necessary and sufficient to ensure pointwise convergence in such systems?
• RQ5 Can the convergence result for the triple average be deduced from the convergence of simpler averages via factorization and transfinite induction?

Key findings

• For every ergodic measure-preserving system $(X,\mathcal{X},\mu,S,T)$ with commuting $S$ and $T$, the triple average $\frac{1}{N^3}\sum_{i,j,k=0}^{N-1} f_0(S^jT^k x)f_1(S^{i+j}T^k x)f_2(S^jT^{i+k}x)$ converges for $\mu$-a.e. $x$ as $N \to \infty$ for all $f_0,f_1,f_2 \in L^\infty(\mu)$.
• In the case of an ergodic distal system with commuting $S$ and $T$, the simpler average $\frac{1}{N}\sum_{i=0}^{N-1} f_1(S^i x)f_2(T^i x)$ converges for $\mu$-a.e. $x$ as $N \to \infty$ for all $f_1,f_2 \in L^\infty(\mu)$.
• Every such system admits a topological model with a strictly ergodic $N_{S,T}(X)$ structure, enabling the use of continuous functions as a dense algebra.
• The proof relies on constructing a triple magic extension and using transfinite induction over isometric extensions to propagate convergence from base factors.
• The convergence in the distal case is established by reducing the problem to a product system where $S$ and $T$ act identically on the common factor, allowing application of the Birkhoff Ergodic Theorem.
• The key technical innovation is the use of topological models with controlled structure to handle the joint dynamics of two commuting transformations in pointwise ergodic theory.