Small scale quantum ergodicity in cat maps. I

TL;DR

This paper establishes quantum ergodicity at small scales for linear hyperbolic maps of the torus, specifically for cat maps, demonstrating ergodic behavior at logarithmic and polynomial scales in certain cases.

Contribution

It proves quantum ergodicity at small scales for all integers N, including logarithmic and polynomial scales, in specific subsets and eigenbases.

Findings

Quantum ergodicity at logarithmic scales for all N.

Quantum ergodicity at polynomial scales in special cases.

Results hold for Hecke eigenbasis and full density subsets.

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.