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

TL;DR

This paper reformulates Furstenberg's $ imes 2 imes 3$ conjecture and related statements in complex analysis and operator algebras, providing new equivalent formulations and insights into these longstanding open problems.

Contribution

It introduces nine equivalent conjectures in complex analysis and operator algebras related to Furstenberg's conjecture, linking ergodic theory with these mathematical frameworks.

Findings

Furstenberg's conjecture is equivalent to Carathéodory functions being convex combinations of rational functions.

Operator-algebraic conjectures involve tracial states on group C*-algebras related to Baumslag-Solitar groups.

Provides nine grouped conjectures connecting ergodic theory, complex analysis, and operator algebras.

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 Carath\'{e}odory 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.