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Published April 29, 2021 | Submitted
Journal Article Open

Rigidity in dynamics and Möbius disjointness


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 ≥ 2, for C^(2+ϵ)-smooth skew products over rotations and C^(2+ϵ)-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.

Additional Information

© 2021 Instytut Matematyczny. MR acknowledges the partial support of a Sloan fellowship and NSF grant DMS-1902063. We would like to thank the American Institute of Mathematics for hosting a workshop on "Sarnak's Conjecture" at which this work was begun. We are grateful to Sacha Mangerel and Joni Teräväinen for bringing to our attention an issue in the proof of Theorem 3.1 in the previous manuscript and to Krzysztof Fra̧czek for discussions on Theorem 4.1.

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