Entropy theory for sectional hyperbolic flows

José Pacifico, Maria; Yang, Fan; Yang, Jiagang

doi:10.1016/j.anihpc.2020.10.001

José Pacifico, Maria ¹ ; Yang, Fan ² ; Yang, Jiagang ^3,⁴

¹ a Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil
² b Department of Mathematics, University of Oklahoma, Norman, OK, USA
³ c Department of Mathematics, Southern University of Science and Technology of China, Guangdong, China
⁴ d Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil

Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1001-1030.

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Résumé

We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for $C^{1}$ flows, every sectional hyperbolic set Λ is entropy expansive, and the topological entropy varies continuously with the flow. Furthermore, if Λ is Lyapunov stable, then it has positive entropy; in addition, if Λ is a chain recurrent class, then it contains a periodic orbit. As a corollary, we prove that for $C^{1}$ generic flows, every Lorenz-like class is an attractor.

Reçu le : 2019-06-27
Révisé le : 2020-09-10
Accepté le : 2020-10-05

MR Zbl

DOI : 10.1016/j.anihpc.2020.10.001

Mots-clés : Entropy, Entropy expansive, Lorenz like attractors, Sectional hyperbolic flows

Affiliations des auteurs :

José Pacifico, Maria ¹ ; Yang, Fan ² ; Yang, Jiagang ^{3, 4}

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José Pacifico, Maria; Yang, Fan; Yang, Jiagang. Entropy theory for sectional hyperbolic flows. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1001-1030. doi : 10.1016/j.anihpc.2020.10.001. http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.001/

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Cité par Sources :

M.J.P. and J.Y. are partially supported by Grant Produtividade em Pesquisa ( CNPq ), Grant Cientista do Nosso Estado ( FAPERJ ), PROEX-CAPES . F.Y. would like to thank the hospitality of Southern University of Science and Technology of China (SUSTC), where part of this work is done.