Unveiling universality, encloseness, and orthogonality in dynamics

TL;DR

This paper explores the universality and orthogonality properties in dynamical systems, connecting them to Sarnak's conjecture and measure-theoretic automorphisms, with implications for orthogonality phenomena.

Contribution

It establishes the existence of universal topological models for certain classes of measure-preserving automorphisms and links these models to orthogonality results related to Sarnak's conjecture.

Findings

Automorphisms with relative discrete spectrum admit universal models.

Several classes including weakly mixing automorphisms have universal models.

Zero entropy systems with countable eigenvalues satisfy Sarnak's conjecture along a subsequence.

Abstract

Motivated by Sarnak's conjecture on M\"obius 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.