{"type":"journal_article","page":"1-58","issue":"1","publication":"Ergodic Theory and Dynamical Systems","date_created":"2022-05-17T12:55:26Z","status":"public","author":[{"full_name":"ARNOLDI, JEAN FRANCOIS","last_name":"ARNOLDI","first_name":"JEAN FRANCOIS"},{"last_name":"FAURE","full_name":"FAURE, FRÉDÉRIC","first_name":"FRÉDÉRIC"},{"full_name":"Weich, Tobias","last_name":"Weich","id":"49178","first_name":"Tobias","orcid":"0000-0002-9648-6919"}],"publication_identifier":{"issn":["0143-3857","1469-4417"]},"intvolume":"        37","_id":"31291","year":"2015","doi":"10.1017/etds.2015.34","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>"}],"volume":37,"keyword":["Applied Mathematics","General Mathematics"],"title":"Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps","external_id":{"arxiv":["1302.3087"]},"publication_status":"published","date_updated":"2022-05-19T10:15:54Z","language":[{"iso":"eng"}],"department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"publisher":"Cambridge University Press (CUP)","citation":{"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>.","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>","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>.","short":"J.F. ARNOLDI, F. FAURE, T. Weich, Ergodic Theory and Dynamical Systems 37 (2015) 1–58.","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} }","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>","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>."},"user_id":"49178"}