Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

TL;DR

This paper studies the spectral properties of open partially expanding maps with fractal trapped sets, establishing an asymptotic spectral gap and a fractal Weyl law for Ruelle resonances, supported by numerical simulations.

Contribution

It introduces the 'minimal captivity' hypothesis, proving the existence of a spectral gap and a fractal Weyl law for Ruelle resonances in this setting.

Findings

Existence of a well-defined discrete spectrum of Ruelle resonances.

Proof of an asymptotic spectral gap under the minimal captivity hypothesis.

Numerical evidence supporting the theoretical results.

Abstract

We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotics of correlation functions for large time and which is parametrized by the Fourier component \nu on the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existence of an asymptotic spectral gap and a Fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit \nu -> infinity. Some numerical computations with the truncated Gauss map illustrate these results.