Rigidity in dynamics and M

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 ≥ 2, for C 2+ -smooth skew products over rotations and for C 2+ -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.

Disjointness of Möbius from asymptotically periodic functions

“…The major ingredient of our proof of the above result is the estimate [28] on averages of multiplicative functions in short intervals and the large sieve. We defer the proof to Appendix C. Here we list some recent results on averages of multiplicative functions in short arithmetic progressions: the method used in [29] can give that the coefficient before log log h log h is k in formula (34); [23,Theorem 3.1] gave the result that when f = μ(n), then for any > 0, the left hand side of formula (34)≤ h 2 Xϕ(k), whenever p|k 1/p ≤ (1 − ) p≤h 1/p; For general multiplicative function, [24, Theorem 1.6, Corollary 1.7] gave the result that for any > 0, the left hand side of formula (34)≤ h 2 Xϕ(k) when k is h 2 -typical (i.e., there are not many prime factors of k less than h 2 ). The reason that we give the estimate in form of formula (34) is that our main interest is to concern about when k is far larger than h, whether the first term of the right hand side of formula (34) is still h 2 Xϕ(k)o h (1).…”

“…In recent years, a lot of progress have been made on Conjecture 1. See [2,3,11,12,18,19,20,23,26,27,33,37,39,42,43], to list a few. In the following, we shall discuss only the results that are more related to this paper.…”

Sarnak's Conjecture from the Ergodic Theory Point of View

“…Rigid systems. In [39], Kanigowski, Lemańczyk and Radziwiłł study rigid systems. 11 To formulate their results, we need some definitions and facts.…”

“…A key tool here is a strengthening of the main result of Matomäki and Radziwiłł [47] (cf. ( 12)) to short interval behaviour along arithmetic progressions: Theorem 6.7 ( [39]). For each ε > 0, there exists L 0 such that for each L ≥ L 0 and q ≥ 1 satisfying p|q 1/p ≤ (1 − ε) p≤L 1/p we can find M 0 = M 0 (q, L) such that for all M ≥ M 0 , we have M/Lq j=0 q−1 a=0 m∈[z+jLq,z+(j+1)Lq] m≡a mod q µ(m) < εM 11 A measure-theoretic system (R, Z, C , κ) is rigid if, for some increasing sequence (qn) of natural numbers, we have f • R qn → f in L 2 (Z, κ) for each f ∈ L 2 (Z, κ).…”

“…This, in particular, applies to all ergodic transformations with discrete spectrum. Moreover, it is shown in [39] that for a.e. IET (of d ≥ 3 intervals) BPV rigidity holds, so a.e.…”