Entropie polynomiale des homéomorphismes de Brouwer
Hauseux, Louis1 ; Le Roux, Frédéric2
1 101 Ter rue des Sources 92160 Antony (France)
2 Institut de Mathématiques de Jussieu - Paris Rive Gauche Sorbonne Université - Campus Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 (France)
Annales Henri Lebesgue, Tome 2 (2019), pp. 39-57.

Nous nous proposons d’étudier l’entropie polynomiale de la composante errante de n’importe quel système dynamique topologique inversible. Pour illustrer cette étude, nous calculerons l’entropie polynomiale de divers homéomorphismes de Brouwer, qui sont les homéomorphismes du plan sans point fixe et préservant l’orientation. En particulier, nous verrons que l’entropie polynomiale de tels homéomorphismes peut prendre n’importe quelle valeur supérieure ou égale à 2.

We study the polynomial entropy of the wandering part of any invertible dynamical system on a compact metric space. As an application we compute the polynomial entropy of Brouwer homeomorphisms (fixed point free orientation preserving homeomorphisms of the plane), and show in particular that it takes every real value greater or equal to 2. (An english version of this text may be found on ArXiv [HLR17])

Reçu le :
Révisé le :
Accepté le :
Première publication :
Publié le :
DOI : 10.5802/ahl.12
Classification : 37E30, 37B40
Mots clés : polynomial entropy, Brouwer homeomorphisms, wandering set
Hauseux, Louis 1 ; Le Roux, Frédéric 2

1 101 Ter rue des Sources 92160 Antony (France)
2 Institut de Mathématiques de Jussieu - Paris Rive Gauche Sorbonne Université - Campus Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 (France) 
@article{AHL_2019__2__39_0,
     author = {Hauseux, Louis and Le Roux, Fr\'ed\'eric},
     title = {Entropie polynomiale des hom\'eomorphismes de {Brouwer}},
     journal = {Annales Henri Lebesgue},
     pages = {39--57},
     publisher = {\'ENS Rennes},
     volume = {2},
     year = {2019},
     doi = {10.5802/ahl.12},
     zbl = {1423.37044},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/ahl.12/}
}
TY  - JOUR
AU  - Hauseux, Louis
AU  - Le Roux, Frédéric
TI  - Entropie polynomiale des homéomorphismes de Brouwer
JO  - Annales Henri Lebesgue
PY  - 2019
SP  - 39
EP  - 57
VL  - 2
PB  - ÉNS Rennes
UR  - http://archive.numdam.org/articles/10.5802/ahl.12/
DO  - 10.5802/ahl.12
LA  - fr
ID  - AHL_2019__2__39_0
ER  - 
%0 Journal Article
%A Hauseux, Louis
%A Le Roux, Frédéric
%T Entropie polynomiale des homéomorphismes de Brouwer
%J Annales Henri Lebesgue
%D 2019
%P 39-57
%V 2
%I ÉNS Rennes
%U http://archive.numdam.org/articles/10.5802/ahl.12/
%R 10.5802/ahl.12
%G fr
%F AHL_2019__2__39_0
Hauseux, Louis; Le Roux, Frédéric. Entropie polynomiale des homéomorphismes de Brouwer. Annales Henri Lebesgue, Tome 2 (2019), pp. 39-57. doi : 10.5802/ahl.12. http://archive.numdam.org/articles/10.5802/ahl.12/

[BL16] Bernard, Patrick; Labrousse, Clémence An entropic characterization of the flat metrics on the two torus, Geom. Dedicata, Volume 180 (2016), pp. 187-201 | DOI | MR | Zbl

[BLR03] Béguin, François; Le Roux, Frédéric Ensemble oscillant d’un homéomorphisme de Brouwer, homéomorphismes de Reeb, Bull. Soc. Math. Fr., Volume 131 (2003) no. 2, pp. 149-210 | DOI | MR | Zbl

[CK97] Cassaigne, Julien; Karhumäki, Juhani Toeplitz words, generalized periodicity and periodically iterated morphisms, Eur. J. Comb., Volume 18 (1997) no. 5, pp. 497-510 | DOI | MR | Zbl

[HLR17] Hauseux, Louis; Le Roux, Frédéric Polynomial entropy of Brouwer homeomorphisms (2017) (https://arxiv.org/abs/1712.01502) | Zbl

[Kan18] Kanigowski, Adam Slow entropy for some smooth flows on surfaces, Isr. J. Math., Volume 226 (2018) no. 2, pp. 535-577 | DOI | MR | Zbl

[KT97] Katok, Anatole; Thouvenot, Jean-Paul Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 33 (1997) no. 3, pp. 323-338 | DOI | Numdam | MR | Zbl

[Kus67] Kushnirenko, A. G. Metric invariants of entropy type, Usp. Mat. Nauk, Volume 22 (1967) no. 5 (137), pp. 57-65 | MR | Zbl

[KVW18] Kanigowski, Adam; Vinhage, Kurt; Wei, Daren Kakutani Equivalence of Unipotent Flows (2018) (https://arxiv.org/abs/1805.01501)

[Lab13] Labrousse, Clémence Polynomial entropy for the circle homeomorphisms and for C 1 nonvanishing vector fields on 𝕋 2 (2013) (https://arxiv.org/abs/1311.0213)

[LR99] Le Roux, Frédéric Bounded recurrent sets for planar homeomorphisms, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 4, pp. 1085-1091 | DOI | MR | Zbl

[Mar13] Marco, Jean-Pierre Polynomial entropies and integrable Hamiltonian systems, Regul. Chaotic Dyn., Volume 18 (2013) no. 6, pp. 623-655 | DOI | MR | Zbl

[Nak95a] Nakayama, Hiromichi A non-flowable plane homeomorphism whose non-Hausdorff set consists of two disjoint lines, Houston J. Math., Volume 21 (1995) no. 3, pp. 569-572 | MR | Zbl

[Nak95b] Nakayama, Hiromichi On dimensions of non-Hausdorff sets for plane homeomorphisms, J. Math. Soc. Japan, Volume 47 (1995) no. 4, pp. 789-793 | DOI | MR | Zbl

[Wal82] Walters, Peter An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer, 1982, ix+250 pages | MR | Zbl

Cité par Sources :