Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy

Abstract

We show that positive Lyapunov exponents imply upper quantum dynamical bounds for Schrödinger operators $H_{f,\theta }u\left(n\right)=u\left(n+1\right)+u\left(n-1\right)+\varphi \left(f^n\theta \right)u\left(n\right)$, where $\varphi :\mathcal{ℳ}\to ℝ$ is a piecewise Hölder function on a compact Riemannian manifold $\mathcal{ℳ}$, and $f:\mathcal{ℳ}\to \mathcal{ℳ}$ is a uniquely ergodic volume-preserving map with zero topological entropy. As corollaries we also obtain localization-type statements for shifts and skew-shifts on higher-dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multifrequency quasiperiodic and skew-shift potentials.