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Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras

Peter Burton, Jane Panangaden

Abstract

Furstenberg's $\times 2 \times 3$ conjecture has remained a central open problem in ergodic theory for over $50$ years, and it serves as the basic test case for a broad class of rigidity phenomena which are believed to hold in number-theoretic dynamics. More recently, two related statements have appeared in the literature: a question about periodic approximation raised by Levit and Vigdorovich in the context of approximate group theory and a periodic equidistribution conjecture formulated by Lindenstrauss. The purpose of this article is to provide equivalent formulations for these three statements in a complex-analytic setting and an operator-algebraic setting, giving nine conjectures grouped into three triples. The complex-analytic conjectures involve so-called Carathéodory functions on the unit disk that satisfy a certain functional identity, and we find that Furstenberg's conjecture is equivalent to the assertion that every such function is a convex combination of rational functions. The operator-algebraic conjectures involve tracial states on the full group $C^\ast$-algebra of a certain semidirect product, which is related to Baumslag-Solitar groups.

Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras

Abstract

Furstenberg's conjecture has remained a central open problem in ergodic theory for over years, and it serves as the basic test case for a broad class of rigidity phenomena which are believed to hold in number-theoretic dynamics. More recently, two related statements have appeared in the literature: a question about periodic approximation raised by Levit and Vigdorovich in the context of approximate group theory and a periodic equidistribution conjecture formulated by Lindenstrauss. The purpose of this article is to provide equivalent formulations for these three statements in a complex-analytic setting and an operator-algebraic setting, giving nine conjectures grouped into three triples. The complex-analytic conjectures involve so-called Carathéodory functions on the unit disk that satisfy a certain functional identity, and we find that Furstenberg's conjecture is equivalent to the assertion that every such function is a convex combination of rational functions. The operator-algebraic conjectures involve tracial states on the full group -algebra of a certain semidirect product, which is related to Baumslag-Solitar groups.

Paper Structure

This paper contains 54 sections, 31 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Nine conjectures
  3. Ergodic theory on the circle
  4. Carathéodory functions
  5. Tracial states on a number-theoretic $C^\ast$-algebra
  6. Statement of results
  7. Measure-function correspondence
  8. Introducing the Herglotz correspondence
  9. Measures to functions
  10. Functions to measures
  11. Exchange of Fourier coefficients and Taylor coefficients
  12. (Proof of Clause I)
  13. (Proof of Clause II)
  14. Homeomorphic property of the correspondence
  15. Preliminary estimates
  16. ...and 39 more sections

Key Result

Theorem 1.1

Conjectures conj.erg-1, conj.comp-1 and conj.see-1 are equivalent.

Theorems & Definitions (73)

  • Definition 1.1
  • Conjecture 1.1: Furstenberg's $\times (2,3)$ conjecture
  • Remark 1.1
  • Conjecture 1.2: Periodic approximation conjecture, after Levit and Vigdorovich
  • Conjecture 1.3: Bourgain-Lindenstrauss periodic equidistribution conjecture
  • Definition 1.2
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Definition 1.3
  • ...and 63 more