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

Published online by Cambridge University Press:  07 October 2015

JEAN FRANCOIS ARNOLDI ,
FRÉDÉRIC FAURE  and
TOBIAS WEICH
JEAN FRANCOIS ARNOLDI
Affiliation:
Centre for Biodiversity Theory and Modelling, Station d’Ecologie Expérimentale du CRNRS, 09200 Moulis, France email arnoldi.jeff@gmail.com
FRÉDÉRIC FAURE
Affiliation:
Institut Fourier, UMR 5582, 100 rue des Maths, BP74 38402 St Martin d’Hères, France email frederic.faure@ujf-grenoble.fr
TOBIAS WEICH
Affiliation:
Fachbereich Mathematik, Philipps-Universität Marburg, a Hans-Meerwein-Straße, 35032 Marburg, Germany email weich@mathematik.uni-marburg.de
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Abstract

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in 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 in a neighborhood of ${\mathcal{K}}$. 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 $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 1 - 58
Copyright
© Cambridge University Press, 2015 

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