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Rigidity in dynamics and Möbius disjointness

Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł

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 $C^{2+ε}$-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.

Rigidity in dynamics and Möbius disjointness

Abstract

Let be a topological dynamical system. We show that if each invariant measure of 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 , then 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 intervals with , for -smooth skew products over rotations and -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.

Paper Structure

This paper contains 12 sections, 20 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Completely rigid points in topological dynamics
  3. Consequences for multiplicative functions
  4. Properties of Möbius function and the proof of Theorem \ref{['th:main1']}
  5. Möbius function on short intervals along arithmetic progressions
  6. A short discussion of BPV and PR rigidity
  7. Interval exchange transformations
  8. Anzai skew products
  9. Smooth flows on ${\mathbb{T}}^2$ and Rokhlin extensions
  10. Proof of Theorem \ref{['th:main1']}
  11. Systems with countably many ergodic rigid measures. Proof of Theorem \ref{['thm:mainE']}
  12. Appendix: Proof of Theorem \ref{['t:main2']}

Key Result

Theorem 1.1

Let $(X,T)$ be a good topological dynamical system. Then $(X,T)$ is Möbius disjoint, that is, sar1 holds for all continuous $f : X \to {\mathbb{R}}$ and all $x \in X$.

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Remark 2.2
  • ...and 28 more