---
_id: '31291'
abstract:
- lang: eng
  text: <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>
author:
- first_name: JEAN FRANCOIS
  full_name: ARNOLDI, JEAN FRANCOIS
  last_name: ARNOLDI
- first_name: FRÉDÉRIC
  full_name: FAURE, FRÉDÉRIC
  last_name: FAURE
- first_name: Tobias
  full_name: Weich, Tobias
  id: '49178'
  last_name: Weich
  orcid: 0000-0002-9648-6919
citation:
  ama: ARNOLDI JF, FAURE F, Weich T. Asymptotic spectral gap and Weyl law for Ruelle
    resonances of open partially expanding maps. <i>Ergodic Theory and Dynamical Systems</i>.
    2015;37(1):1-58. doi:<a href="https://doi.org/10.1017/etds.2015.34">10.1017/etds.2015.34</a>
  apa: ARNOLDI, J. F., FAURE, F., &#38; Weich, T. (2015). Asymptotic spectral gap
    and Weyl law for Ruelle resonances of open partially expanding maps. <i>Ergodic
    Theory and Dynamical Systems</i>, <i>37</i>(1), 1–58. <a href="https://doi.org/10.1017/etds.2015.34">https://doi.org/10.1017/etds.2015.34</a>
  bibtex: '@article{ARNOLDI_FAURE_Weich_2015, title={Asymptotic spectral gap and Weyl
    law for Ruelle resonances of open partially expanding maps}, volume={37}, DOI={<a
    href="https://doi.org/10.1017/etds.2015.34">10.1017/etds.2015.34</a>}, 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} }'
  chicago: 'ARNOLDI, JEAN FRANCOIS, FRÉDÉRIC FAURE, and Tobias Weich. “Asymptotic
    Spectral Gap and Weyl Law for Ruelle Resonances of Open Partially Expanding Maps.”
    <i>Ergodic Theory and Dynamical Systems</i> 37, no. 1 (2015): 1–58. <a href="https://doi.org/10.1017/etds.2015.34">https://doi.org/10.1017/etds.2015.34</a>.'
  ieee: 'J. F. ARNOLDI, F. FAURE, and T. Weich, “Asymptotic spectral gap and Weyl
    law for Ruelle resonances of open partially expanding maps,” <i>Ergodic Theory
    and Dynamical Systems</i>, vol. 37, no. 1, pp. 1–58, 2015, doi: <a href="https://doi.org/10.1017/etds.2015.34">10.1017/etds.2015.34</a>.'
  mla: ARNOLDI, JEAN FRANCOIS, et al. “Asymptotic Spectral Gap and Weyl Law for Ruelle
    Resonances of Open Partially Expanding Maps.” <i>Ergodic Theory and Dynamical
    Systems</i>, vol. 37, no. 1, Cambridge University Press (CUP), 2015, pp. 1–58,
    doi:<a href="https://doi.org/10.1017/etds.2015.34">10.1017/etds.2015.34</a>.
  short: J.F. ARNOLDI, F. FAURE, T. Weich, Ergodic Theory and Dynamical Systems 37
    (2015) 1–58.
date_created: 2022-05-17T12:55:26Z
date_updated: 2022-05-19T10:15:54Z
department:
- _id: '10'
- _id: '623'
- _id: '548'
doi: 10.1017/etds.2015.34
external_id:
  arxiv:
  - '1302.3087'
intvolume: '        37'
issue: '1'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 1-58
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  issn:
  - 0143-3857
  - 1469-4417
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially
  expanding maps
type: journal_article
user_id: '49178'
volume: 37
year: '2015'
...