2019
DOI: 10.48550/arxiv.1905.13256
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Rigidity in dynamics and Möbius disjointness

Abstract: Let (X, T ) be a topological dynamical system. We show that if all invariant measures of (X, T ) give rise to measure theoretic dynamical system that are rigid 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 they all give rise to rigid measure theoretic dynamical systems. This recovers several earlier results and immediately implies Sarnak's conjecture in the following new cases: for almos… Show more

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Cited by 10 publications
(29 citation statements)
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“…Applying Theorem 1.1, we obtain a result on the disjointness of µ(n)e(P (n)) from certain rigid systems. This is closely related to a recent result established by Kanigowski, Lemańczyk and Radziwi l l in [KLR19] on the Möbius disjointness of certain rigid systems.…”
Section: Introductionsupporting
confidence: 82%
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“…Applying Theorem 1.1, we obtain a result on the disjointness of µ(n)e(P (n)) from certain rigid systems. This is closely related to a recent result established by Kanigowski, Lemańczyk and Radziwi l l in [KLR19] on the Möbius disjointness of certain rigid systems.…”
Section: Introductionsupporting
confidence: 82%
“…In [KLR19], Kanigowski, Lemańczyk and Radziwi l l showed that the average of µ(n) is small on almost all arithmetic progressions a(mod s) and almost all short intervals [n + 1, n + hs]. 1 In fact, they showed that for any s ≥ 1 and h ≥ 3, lim sup where ϕ is the Euler totient function.…”
Section: Introductionmentioning
confidence: 99%
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“…for some g : T 1 → R, i..e., T (α,A) has the form (x, ϕ) → (x + α, ϕ + g(x)), then the action T (α,A) is known to satisfy the Möbius disjointness conjecture as long as g is C 1+ǫ by the recent work of de Faveri [16] (see also the earlier works [25,29,31,34,42], which assumed higher regularity and/or other conditions).…”
Section: Introductionmentioning
confidence: 99%