Measure complexity and Möbius disjointness

Abstract

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity.Moreover, it is proved that the following classes of topological dynamical systems (X, T ) meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of T has discrete spectrum.(2) T is a homotopically trivial C ∞ skew product system on T 2 over an irrational rotation of the circle. Combining this with the previous results it implies that the Möbius disjointness conjecture holds for any C ∞ skew product system on T 2 . (3) T is a continuous skew product map of the form (ag, y + h(g)) on G × T 1 over a minimal rotation of the compact metric abelian group G and T preserves a measurable section. (4) T is a tame system.

Bounded complexity, mean equicontinuity and discrete spectrum

“…It is conjectured in [15] the converse of Proposition 4.5 is also true. If µ is ergodic, by [12, Corollary 39], we know that µ has discrete spectrum and if and only if µ is mean equicontinuous.…”

“…In [18] Katok introduced a notion using the modified notion of spanning sets with respect to an invariant measure µ and an error ε, which can be used to define the complexity function. In [2] Recently, in the investigation of the Sanark conjecture, Huang, Wang and Ye [15] introduced the measure complexity of an invariant measure µ similar to the one introduced by Katok [18], by using the mean metric instead of the Bowen metric (for discussion and results related to mean metric, see also [23,30]). They showed that if an invariant measure has discrete spectrum, then the measure complexity with respect to this invariant measure is bounded.…”

Measure-theoretic mean equicontinuity and bounded complexity

“…(X , T ) and showed that (X , T, µ) is µ-mean equicontinuous if and only if it has discrete spectrum, see also [20] for another proof. It should be noticed that recently the authors in [14,17] showed that for an invariant measure µ on a t.d.s. (X , T ), (X , T, µ) is µ-mean equicontinuous if and only if it has discrete spectrum.…”

“…In [3], Blanchard et al studied topological complexity via the complexity function of an open cover and showed that the complexity function is bounded for any open cover if and only if the system is equicontinuous. In [14,17], Huang et al studied topological and measure-theoretic complexity via a sequence of metrics induced by a metric and showed that an invariant measure µ on (X , T ) has bounded complexity with respect tod n if and only if (X , T ) is µ-mean equicontinuous if and only if it has discrete spectrum, whered n (x, y) = 1 n ∑ n−1 i=0 d (T i x, T i y). Ferenczi [5] studied measure-theoretic complexity of a m.p.s (X , B, µ, T ) using αnames of a partition and the Hamming distance.…”

Sarnak’s Conjecture: What’s New

“…for each ε ą 0 (here d n py, zq " 1 n ř n j"1 dpT j y, T j zq). The main result of the recent article [90] states the following:…”