Small scale quantum ergodicity in cat maps. I

Abstract

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck constant. First, for all integers $N\in \mathbb{N}$, we show quantum ergodicity at logarithmical scales $|\log h{|}^{-\alpha }$ for some $\alpha >0$. Second, we show quantum ergodicity at polynomial scales $h^{\alpha }$ for some $\alpha >0$, in two special cases: $N\in S(\mathbb{N})$ of a full density subset $S(\mathbb{N})$ of integers and Hecke eigenbasis for all integers.

20 pages