[Propriétés génériques pour des systèmes dynamiques de faible régularité]
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 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 -regularity and Lipschitz regularity, focussing on two cases: Hölder mappings and Sobolev mappings.
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@article{CRMATH_2017__355_11_1185_0,
author = {de Faria, Edson and Hazard, Peter and Tresser, Charles},
title = {Infinite entropy is generic in {H\"older} and {Sobolev} spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {1185--1189},
publisher = {Elsevier},
volume = {355},
number = {11},
year = {2017},
doi = {10.1016/j.crma.2017.10.016},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.10.016/}
}
TY - JOUR AU - de Faria, Edson AU - Hazard, Peter AU - Tresser, Charles TI - Infinite entropy is generic in Hölder and Sobolev spaces JO - Comptes Rendus. Mathématique PY - 2017 SP - 1185 EP - 1189 VL - 355 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.10.016/ DO - 10.1016/j.crma.2017.10.016 LA - en ID - CRMATH_2017__355_11_1185_0 ER -
%0 Journal Article %A de Faria, Edson %A Hazard, Peter %A Tresser, Charles %T Infinite entropy is generic in Hölder and Sobolev spaces %J Comptes Rendus. Mathématique %D 2017 %P 1185-1189 %V 355 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.10.016/ %R 10.1016/j.crma.2017.10.016 %G en %F CRMATH_2017__355_11_1185_0
de Faria, Edson; Hazard, Peter; Tresser, Charles. Infinite entropy is generic in Hölder and Sobolev spaces. Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1185-1189. doi : 10.1016/j.crma.2017.10.016. http://www.numdam.org/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.
