Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras
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| Autores principales: | , |
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| Formato: | Preprint |
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2024
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| _version_ | 1866914997747056640 |
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| author | Burton, Peter Panangaden, Jane |
| author_facet | Burton, Peter Panangaden, Jane |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras Burton, Peter Panangaden, Jane Dynamical Systems Complex Variables Operator Algebras 37A44, 37A55, 30E20, 46L55 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. |
| title | Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras |
| topic | Dynamical Systems Complex Variables Operator Algebras 37A44, 37A55, 30E20, 46L55 |
| url | https://arxiv.org/abs/2410.22701 |