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}$isapiecewiseH\ddot{o}lderfunctiononacompactRiemannianmanifold$$\mathcal{M}$$,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.