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

Abstract

We consider a specific family of skew product 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 Tsujii (2008 Ergodic Theory Dyn. Syst. 28 291–317). 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.