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

In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schr\"odinger operators $H_{f,\theta} u(n)=u(n+1)+u(n-1)+ \phi(f^n\theta)u(n)$, where $\phi : \mathcal{M}\to {\Bbb R}$ is a piecewise H\"older 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 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 multi-frequency quasiperiodic and skew-shift potentials.