Joint ergodicity - 40 years on

TL;DR

This survey reviews 40 years of progress in joint ergodicity, focusing on recent advances, new tools, open problems, and challenges in understanding the convergence of multiple ergodic averages.

Contribution

It summarizes recent developments and introduces new tools in the study of joint ergodicity, highlighting open problems and current challenges in the field.

Findings

Summary of 40 years of progress in joint ergodicity

Introduction of new analytical tools for multiple ergodic averages

Identification of key open problems and challenges

Abstract

Recent years have seen dramatic progress in the study of joint ergodicity, i.e. a scenario in which a multiple ergodic average converges in norm to the product of integrals of individual functions. This survey, accompanying the talk given by the author in the Perspectives on Ergodic Theory and its Interactions conference to celebrate Vitaly Bergelson's 75th birthday, aims to summarize these recent advances, outline crucial new tools, present various open problems, and highlight the main challenges currently faced in the study of multiple ergodic averages.