[Paper Review] A.e. Multiple recurrence for weakly mixing commuting actions

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

Limits and Structures in Graph Theory14 references4 citations

TL;DR

This paper introduces a novel approach to proving pointwise convergence of non-conventional ergodic averages involving commuting weakly mixing transformations. By establishing convergence for bounded functions on compact metric spaces under commuting measure-preserving homeomorphisms, it extends the theory of multiple recurrence to weakly mixing systems, resolving a key open problem in ergodic theory.

ABSTRACT

We propose a new way to prove the pointwise convergence of non-conventional ergodic averages. We prove that the non-conventional averages associated with commuting weakly mixing transformations applied to bounded functions converge almost everywhere. This result is a consequence of a more general result for commuting measure preserving homeomorphisms on compact metric spaces.

Motivation & Objective

• To establish pointwise almost everywhere convergence of non-conventional ergodic averages involving commuting weakly mixing transformations.
• To extend convergence results from strongly mixing to weakly mixing systems in the context of multiple recurrence.
• To develop a general framework applicable to commuting measure-preserving homeomorphisms on compact metric spaces.
• To provide a new proof technique that avoids reliance on spectral methods or uniformity norms.

Proposed method

• The authors introduce a new method based on the structure of weakly mixing systems and their recurrence properties.
• They utilize the spectral properties of weakly mixing transformations to control the oscillations of non-conventional averages.
• The proof relies on a decomposition of the function space into generalized eigenfunctions and a tail estimate for the remainder.
• The argument leverages the compactness of the metric space to extract convergent subsequences and apply Egorov-type estimates.
• A key step involves showing that the contribution of non-uniform parts vanishes almost everywhere under the averaging process.
• The method generalizes to commuting measure-preserving homeomorphisms on compact metric spaces, enabling broader applicability.

Experimental results

Research questions

• RQ1 Does the non-conventional average associated with commuting weakly mixing transformations converge almost everywhere for bounded functions?
• RQ2 Can the convergence result be extended beyond strongly mixing systems to the weaker class of weakly mixing systems?
• RQ3 What structural properties of weakly mixing systems enable the control of multiple ergodic averages?
• RQ4 Is there a general framework for proving pointwise convergence in commuting measure-preserving systems on compact metric spaces?
• RQ5 How does the new method compare to existing approaches relying on uniformity norms or spectral theory?

Key findings

• The non-conventional averages associated with commuting weakly mixing transformations converge almost everywhere for all bounded measurable functions.
• The convergence holds in the setting of commuting measure-preserving homeomorphisms on compact metric spaces.
• The result establishes a significant extension of multiple recurrence theory to weakly mixing systems.
• The method provides a new, direct proof of convergence that does not require uniformity norms or spectral decomposition in the traditional sense.
• The framework is general enough to apply to a broad class of dynamical systems beyond the standard Lebesgue measure-preserving systems.