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

J.F. ARNOLDI, F. FAURE, T. Weich, Ergodic Theory and Dynamical Systems 37 (2015) 1–58.

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Journal Article | Published | English
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ARNOLDI, JEAN FRANCOIS; FAURE, FRÉDÉRIC; Weich, TobiasLibreCat
Abstract
<jats:p>We consider a simple model of an open partially expanding map. Its trapped set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline1" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline2" /><jats:tex-math>$\unicode[STIX]{x1D708}$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline3" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline4" /><jats:tex-math>$\unicode[STIX]{x1D708}\rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula>. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.</jats:p>
Keywords
Publishing Year
Journal Title
Ergodic Theory and Dynamical Systems
Volume
37
Issue
1
Page
1-58
ISSN
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### Cite this

ARNOLDI JF, FAURE F, Weich T. Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps. Ergodic Theory and Dynamical Systems. 2015;37(1):1-58. doi:10.1017/etds.2015.34
ARNOLDI, J. F., FAURE, F., & Weich, T. (2015). Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps. Ergodic Theory and Dynamical Systems, 37(1), 1–58. https://doi.org/10.1017/etds.2015.34
@article{ARNOLDI_FAURE_Weich_2015, title={Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps}, volume={37}, DOI={10.1017/etds.2015.34}, number={1}, journal={Ergodic Theory and Dynamical Systems}, publisher={Cambridge University Press (CUP)}, author={ARNOLDI, JEAN FRANCOIS and FAURE, FRÉDÉRIC and Weich, Tobias}, year={2015}, pages={1–58} }
ARNOLDI, JEAN FRANCOIS, FRÉDÉRIC FAURE, and Tobias Weich. “Asymptotic Spectral Gap and Weyl Law for Ruelle Resonances of Open Partially Expanding Maps.” Ergodic Theory and Dynamical Systems 37, no. 1 (2015): 1–58. https://doi.org/10.1017/etds.2015.34.
J. F. ARNOLDI, F. FAURE, and T. Weich, “Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps,” Ergodic Theory and Dynamical Systems, vol. 37, no. 1, pp. 1–58, 2015, doi: 10.1017/etds.2015.34.
ARNOLDI, JEAN FRANCOIS, et al. “Asymptotic Spectral Gap and Weyl Law for Ruelle Resonances of Open Partially Expanding Maps.” Ergodic Theory and Dynamical Systems, vol. 37, no. 1, Cambridge University Press (CUP), 2015, pp. 1–58, doi:10.1017/etds.2015.34.

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arXiv 1302.3087