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

Jean-François Arnoldi (1, 2) , Frédéric Faure (1) , Tobias Weich (3)

1 IF - Institut Fourier
2 SETE - Station d'écologie théorique et expérimentale
3 Fachbereich Physik [Marburg]

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.

Mots clés: Semi-classical analysis.; Semi-classical analysis; Transfer operator; Ruelle resonances; Decay of correlations

Domaines: Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]

Jean-François Arnoldi, Frédéric Faure, Tobias Weich. Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps. Ergodic Theory and Dynamical Systems, 2017, 37 (1), pp.1-58. ⟨10.1017/etds.2015.34⟩. ⟨hal-00787781v2⟩