Semiclassical origin of the spectral gap for transfer operators of partially expanding map

TL;DR

This paper demonstrates that semiclassical analysis reveals a spectral gap in transfer operators of partially expanding maps on the torus, linking the gap to classical dynamics on the cotangent space and a trapped set.

Contribution

It introduces a semiclassical framework to analyze transfer operators, providing a natural description of the spectral gap and its relation to classical dynamics, extending previous results.

Findings

Spectral gap develops in the high-frequency limit for generic maps.

Transfer operator is shown to be a semiclassical operator with classical dynamics.

The spectral gap is linked to the properties of a trapped set in phase space.

Abstract

We consider a simple model of partially expanding map on the torus. We study the spectrum of the Ruelle transfer operator and show that in the limit of high frequencies in the neutral direction (this is a semiclassical limit), the spectrum develops a spectral gap, for a generic map. This result has already been obtained by M. Tsujii (05). The novelty here is that we use semiclassical analysis which provides a different and quite natural description. We show that the transfer operator is a semiclassical operator with a well defined "classical dynamics" on the cotangent space. This classical dynamics has a "trapped set" which is responsible for the Ruelle resonances spectrum. In particular we show that the spectral gap is closely related to a specific dynamical property of this trapped set.