Countably and entropy expansive homeomorphisms with the shadowing property
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- by Alfonso Artigue, Bernardo Carvalho, Welington Cordeiro and José Vieitez PDF
- Proc. Amer. Math. Soc. 150 (2022), 3369-3378 Request permission
Abstract:
We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms $f\colon S\to S$ satisfying the shadowing property and $\Omega (f)=S$, both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism.References
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Additional Information
- Alfonso Artigue
- Affiliation: Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay
- MR Author ID: 863559
- Email: artigue@unorte.edu.uy
- Bernardo Carvalho
- Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais - UFMG Av. Antônio Carlos, 6627 - Campus Pampulha Belo Horizonte - MG, Brazil; and Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Ernst-Abbe-Platz 2 07743 Jena, Germany
- MR Author ID: 1027591
- ORCID: 0000-0002-9400-0882
- Email: bmcarvalho@mat.ufmg.br
- Welington Cordeiro
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich, 8 00-656 Warszawa, Poland
- MR Author ID: 1027592
- Email: wcordeiro@impan.pl
- José Vieitez
- Affiliation: Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay
- Email: jvieitez@unorte.edu.uy
- Received by editor(s): April 28, 2019
- Received by editor(s) in revised form: August 14, 2020
- Published electronically: May 13, 2022
- Additional Notes: The second author was supported by CAPES and the Alexander von Humboldt Foundation under the project number 88881.162174/2017-1 and also by CNPq grant number 405916/2018-3.
Part of this work was developed while the second author was visiting the Departamento de Matemática y Estadística del Litoral in Salto, Uruguay, where some conversations with Mauricio Achigar happened. - Communicated by: Nimish Shah
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3369-3378
- MSC (2020): Primary 37D10; Secondary 37B99
- DOI: https://doi.org/10.1090/proc/15326
- MathSciNet review: 4439460