Unveiling universality, encloseness, and orthogonality in dynamics
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910160345104384 |
|---|---|
| author | Aaronson, J. Danilenko, A. I. Kułaga-Przymus, J. Lemańczyk, M. |
| author_facet | Aaronson, J. Danilenko, A. I. Kułaga-Przymus, J. Lemańczyk, M. |
| contents | Motivated by Sarnak's conjecture on Möbius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation is that whenever a strong form of such orthogonality holds in a system $(X,T)$ then the orthogonality holds for all topological systems in which each ergodic measure yields an automorphism that is measure-theoretically isomorphic to one arising from an ergodic measure in $(X,T)$. This leads us to study two purely dynamical problems: the existence of universal topological models for characteristic classes of measure-preserving automorphisms and the existence of a common ergodic extension for a measurable family of ergodic automorphisms. We show that the class of automorphisms with relative discrete spectrum over the identity factor--as well as several related classes including the weakly mixing case--admit universal models. We also highlight potential applications to the orthogonality phenomena. Moreover, we show that if the set of all measure-theoretic eigenvalues of a zero entropy system $(X,T)$ is countable, then $(X,T)$ satisfies Sarnak's conjecture along a subsequence of full logarithmic density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21392 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unveiling universality, encloseness, and orthogonality in dynamics Aaronson, J. Danilenko, A. I. Kułaga-Przymus, J. Lemańczyk, M. Dynamical Systems 37A05, 37A30, 37A35, 37A44, 37B02, 11K** Motivated by Sarnak's conjecture on Möbius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation is that whenever a strong form of such orthogonality holds in a system $(X,T)$ then the orthogonality holds for all topological systems in which each ergodic measure yields an automorphism that is measure-theoretically isomorphic to one arising from an ergodic measure in $(X,T)$. This leads us to study two purely dynamical problems: the existence of universal topological models for characteristic classes of measure-preserving automorphisms and the existence of a common ergodic extension for a measurable family of ergodic automorphisms. We show that the class of automorphisms with relative discrete spectrum over the identity factor--as well as several related classes including the weakly mixing case--admit universal models. We also highlight potential applications to the orthogonality phenomena. Moreover, we show that if the set of all measure-theoretic eigenvalues of a zero entropy system $(X,T)$ is countable, then $(X,T)$ satisfies Sarnak's conjecture along a subsequence of full logarithmic density. |
| title | Unveiling universality, encloseness, and orthogonality in dynamics |
| topic | Dynamical Systems 37A05, 37A30, 37A35, 37A44, 37B02, 11K** |
| url | https://arxiv.org/abs/2604.21392 |