Comptes Rendus
no. 11
Dynamical systems
Infinite entropy is generic in Hölder and Sobolev spaces
[Propriétés génériques pour des systèmes dynamiques de faible régularité]

Présenté par : Claire Voisin

Edson de Faria1 ; Peter Hazard1 ; Charles Tresser2
1 Instituto de Matemática e Estatística, USP, São Paulo, SP, Brazil
2 Aperio, MATAM Scientific Industrial Ctr., 9 A. Sakharov St., Haïfa, 3508409, Israel
Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1185-1189.

En 1980, Yano a montré que, sur une variété différentielle compacte, pour les endomorphismes en toutes dimensions et les homéomorphismes en dimension plus grande que un, l'entropie topologique est génériquement infinie. Il avait été auparavant montré que, pour les endomorphismes Lipschitz continus, l'entropie est toujours finie. Dans cette note, nous étudions ce qui se passe entre la régularité C0 et la continuité de type Lipschitz, en nous concentrant sur deux cas, endomorphismes et homéomorphismes de classes de Hölder et de Sobolev.

In 1980, Yano showed that on smooth compact manifolds, for endomorphisms in dimension one or above and homeomorphisms in dimensions greater than one, topological entropy is generically infinite. It had earlier been shown that, for Lipschitz endomorphisms on such spaces, topological entropy is always finite. In this article, we investigate what occurs between C0-regularity and Lipschitz regularity, focussing on two cases: Hölder mappings and Sobolev mappings.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.10.016

Edson de Faria 1 ; Peter Hazard 1 ; Charles Tresser 2

1 Instituto de Matemática e Estatística, USP, São Paulo, SP, Brazil
2 Aperio, MATAM Scientific Industrial Ctr., 9 A. Sakharov St., Haïfa, 3508409, Israel 
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Edson de Faria; Peter Hazard; Charles Tresser. Infinite entropy is generic in Hölder and Sobolev spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1185-1189. doi : 10.1016/j.crma.2017.10.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.016/

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Cité par Sources :

☆ This work was partially supported by “Projeto Temático Dinâmica em Baixas Dimensões”, FAPESP Grant no. 2011/16265-2 and 2016/25053-8, FAPESP Grant no. 2015/17909-7, Projeto PVE CNPq 401020/2014-2 and CAPES Grant CSF-PVE-S - 88887.117899/2016-00.

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