Rigidity in dynamics and Möbius disjointness

Abstract

Let $(X,T)$ be a topological dynamical system. We show that if each invariant measure of $(X,T)$ gives rise to a measure-theoretic dynamical system that is either: a. rigid along a sequence of "bounded prime volume" or b. admits a polynomial rate of rigidity on a linearly dense subset in $C(X)$, then $(X,T)$ satisfies Sarnak's conjecture on Möbius disjointness. We show that the same conclusion also holds if there are countably many invariant ergodic measures, and each of them satisfies a. or b. This recovers some earlier results and implies Sarnak's conjecture in the following new cases: for almost every interval exchange map of $d$ intervals with $d\ge 2$, for $C^{2+\epsilon }$-smooth skew products over rotations and $C^{2+\epsilon }$-smooth flows (without fixed points) on the torus. In particular, these are improvements of earlier results of respectively Chaika-Eskin, Wang and Huang-Wang-Ye. We also discuss some purely arithmetic consequences for the Liouville function.

First version had a gap in the proof of Theorem 3.1. The gap is fixed in the current version (at a cost of a slightly weaker statement)

References in corpus (2)

• Multiplicative functions in short arithmetic progressions
• M{ö}bius orthogonality in density for zero entropy dynamical systems

Cited by in corpus (4)

• Disjointness of Möbius from asymptotically periodic functions
• Prime number theorem for regular Toeplitz subshifts
• Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow
• Quantitative almost reducibility and Möbius disjointness for analytic quasiperiodic Schrodinger cocycles