Interf\'erences pour les chats quantiques

TL;DR

This paper investigates quantum dynamics of linear automorphisms on the torus beyond Ehrenfest time, linking wave packet evolution to Birkhoff sums and Diophantine approximation, revealing insights into quantum chaos and automorphism behavior.

Contribution

It introduces a novel approximation of the quantum propagator matrix using Birkhoff sums of nilrotations, connecting quantum dynamics with Diophantine approximation techniques.

Findings

Wave packet basis approximates propagator by Birkhoff sums

Evolved wave packet equidistribution relates to Diophantine approximation

Provides new insights into quantum chaos on the torus

Abstract

In this text I study quantum dynamics of quantized linear automorphisms of the torus after the Ehrenfest time. I show that, in the wave packet basis, the 'matrix' of the associated propagator is well approximated by Birkhoff sums of nilrotations on the torus. In the second part, I conduct a thorough study of these sums and relate the equidistribution of evolved wave packets to a problem of Diophantine approximation.