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Rigidity in dynamics and M

Volume 255 / 2021

Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł Fundamenta Mathematicae 255 (2021), 309-336 MSC: 37A44, 11A25, 11N37. DOI: 10.4064/fm931-11-2020 Published online: 29 April 2021

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 $d \geq 2$, for $C^{2+\epsilon }$-smooth skew products over rotations and for $C^{2+\epsilon }$-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.

Authors

  • Adam KanigowskiDepartment of Mathematics
    University of Maryland at College Park
    College Park, MD 20740, U.S.A.
    e-mail
  • Mariusz LemańczykFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopin St. 12/18
    87-100 Toruń, Poland
    e-mail
  • Maksym RadziwiłłDepartment of Mathematics
    Caltech
    1200 E California Blvd
    Pasadena, CA 91125, U.S.A.
    e-mail

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