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We provide a criterion for a point satisfying the required disjointness condition in Sarnak's M\"obius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with…

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…

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…

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…

We show that there are an irrational rotation $Tx=x+\alpha$ on the circle $\mathbb{T}$ and a continuous $\varphi\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting…

Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A:G\to G $ is a…

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…

We formulate the generalized Sarnak's M\"obius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(\Phi(g(n)\Gamma))_{n\in\mathbb{Z}^{D}}$ and aperiodic…

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 show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy…

We show that all $q$-semimultiplicative sequences are asymptotically orthogonal to the M\"obius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and…

Sarnak's M\"obius disjointness conjecture states that M\"obius function is disjoint to any zero entropy dynamics. We prove that M\"obius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost…

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…

We prove Sarnak's M\"obius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent…

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 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every $f\in…

We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the M\"{o}bius function is a good weight for the ergodic theorem, to maps acting on…

It is shown that the homogeneous ergodic bilinear averages with M\"{o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\mu)$, and $a,b \in \mathbb{Z}$, then for…

Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of…

Let $(X,\nu,T)$ be a measure-preserving system, and let $P_1,\ldots, P_k$ be polynomials with integer coefficients. We prove that, for any $f_1,\ldots, f_k\in L^{\infty}(X)$, the M\"obius-weighted polynomial multiple ergodic averages…