M\"obius disjointness for models of an ergodic system and beyond

TL;DR

This paper investigates the strong MOMO property related to the Möbius function in dynamical systems, establishing conditions under which systems are Möbius disjoint and exploring implications for Sarnak's conjecture and entropy.

Contribution

It proves that the strong MOMO property in uniquely ergodic models implies Möbius disjointness for various classes of systems and links this property to entropy and conjectures like Chowla.

Findings

Uniquely ergodic models of certain systems are Möbius disjoint if they have the strong MOMO property.

Sarnak's conjecture implies uniform Möbius disjointness in zero entropy systems.

The strong MOMO property characterizes zero entropy systems under Chowla's conjecture.

Abstract

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},\kappa,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are M\"obius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbol{\mu}$ in all zero entropy systems, in particular, it makes $\boldsymbol{\mu}$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.