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 functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.

Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.

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We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n\in\mathbb{N}$, under Dirichlet boundary conditions \[ u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad \nu>0, \] where $\eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le C\nu^{-m}, $ uniformy in $t\ge0$ and $\nu>0$. In this work we prove that for small $\nu>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $\nu^{-\kappa_{n,m}}$ with $\kappa_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu})$.

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