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arXiv · 1810.11949

Small scale quantum ergodicity in cat maps. I

Xiaolong Han
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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.

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Xiaolong Han. 2018-10-29. Small scale quantum ergodicity in cat maps. I. https://arxiv.org/abs/1810.11949

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