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Joint ergodicity - 40 years on

Borys Kuca

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.

Joint ergodicity - 40 years on

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.
Paper Structure (68 sections, 89 theorems, 218 equations)

This paper contains 68 sections, 89 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Multiple ergodic averages: what are they, and why should you care?
  3. How to describe the limits of multiple ergodic averages?
  4. Explicit description of a limit
  5. Describing the limit via a characteristic factor or a seminorm
  6. Describing a limit by comparison
  7. Joint ergodicity: what is it, and why is it useful?
  8. What do we study when we study joint ergodicity?
  9. Acknowledgments
  10. Joint ergodicity criteria
  11. Key definitions
  12. Old Joint Ergodicity Strategy
  13. New Joint Ergodicity Strategy
  14. Joint ergodicity for large classes of sequences and systems
  15. Integer polynomials
  16. ...and 53 more sections

Key Result

Theorem 1.1

Let $k,\ell\in{\mathbb N}$. For every $A\subseteq{\mathbb Z}^\ell$ there exists a system $(X, {\mathcal{X}}, \mu,$$T_1, \ldots, T_\ell)$ and a set $E\in{\mathcal{X}}$ with the following properties:

Theorems & Definitions (151)

  • Theorem 1.1: Furstenberg's correspondence principle (cf. Ber87b)
  • Theorem 1.2: Von Neumann's mean ergodic theorem
  • Example 1.3: Local obstructions for polynomials
  • Definition 1.4: Factor and seminorm control
  • Proposition 1.5: Rational Kronecker factor controls single polynomial averages Fu81
  • Example 1.6: Host-Kra theory for Furstenberg averages
  • Theorem 1.7: Host-Kra structure theorem
  • Example 1.8: Host's magic extensions
  • Theorem 1.9: Identity for arithmetic progressions along fractional powers
  • Definition 1.10: Joint ergodicity
  • ...and 141 more