#### Vol. 12, No. 4, 2019

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

### Rui Han and Svetlana Jitomirskaya

Vol. 12 (2019), No. 4, 867–902
DOI: 10.2140/apde.2019.12.867
##### 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 $\varphi :\mathsc{ℳ}\to ℝ$ is a piecewise Hölder function on a compact Riemannian manifold $\mathsc{ℳ}$, and $f:\mathsc{ℳ}\to \mathsc{ℳ}$ 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.

##### Keywords
transport exponent, multifrequency quasiperiodic, skew-shift
##### Mathematical Subject Classification 2010
Primary: 47B36, 81Q10