Unveiling universality, encloseness, and orthogonality in dynamics

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.