Related papers: 0-1 sequences of the Thue-Morse type and Sarnak's …

It is shown that Sarnak's M\"{o}bius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla…

Assuming Sarnak conjecture is true for any singular dynamical process, we prove that the spectral measure of the M\"{o}bius function is equivalent to Lebesgue measure. Conversely, under Elliott conjecture, we establish that the M\"{o}bius…

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's M\"{o}bius disjointness conjecture holds for any system for which every invariant Borel probability measure has…

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms…

Motivated by Sarnak's conjecture on M\"obius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation…

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…

Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the…

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and…

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of…

We construct the counter-example for polynomial version of Sarnak's conjecture for minimal systems, which assets that the M\"obius function is linearly disjoint from subsequences along polynomials of deterministic sequences realized in…

In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ and any $\epsilon>0$, there exists a Toeplitz sequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that the entropy $h({\bf…

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on M\"obius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the M\"obius function…

We show that the M\"obius disjointess of zero entropy dynamical systems implies the existence of an increasing sequence of positive integers along which the Chowla conjecture on autocorrelations of the M\"obius function holds.

The M\"obius disjointness conjecture of Sarnak states that the M\"obius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the…

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic…

In this paper we consider a discrete-time dynamical system on the real line by random iteration of two functions. These functions are assumed to satisfy appropriate monotonicity conditions; optionally, a symmetry condition may be imposed.…

The Thue--Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors $w$ within this sequence, more precisely, the sequence of gaps between consecutive…

We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational…

We show that Sarnak's conjecture on M\"obius disjointness holds for all subshifts given by bijective substitutions and some other similar dynamical systems, e.g.\ those generated by Rudin-Shapiro type sequences.

Sarnak's M\"{o}bius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) \mu (n)= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing…