Hauseux, Louis; Le Roux, Frédéric
Entropie polynomiale des homéomorphismes de Brouwer
Annales Henri Lebesgue, Volume 2  (2019), p. 39-57

Metadata

Keywordspolynomial entropy, Brouwer homeomorphisms, wandering set

Abstract

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.


References

[BL16] Bernard, Patrick; Labrousse, Clémence An entropic characterization of the flat metrics on the two torus, Geom. Dedicata, Volume 180 (2016), pp. 187-201 | Article | MR 3451464

[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 | MR 1988946

[CK97] Cassaigne, Julien; Karhumäki, Juhani Toeplitz words, generalized periodicity and periodically iterated morphisms, Eur. J. Comb., Volume 18 (1997) no. 5, pp. 497-510 | Article | MR 1455183

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

[Kan18] Kanigowski, Adam Slow entropy for some smooth flows on surfaces, Isr. J. Math., Volume 226 (2018) no. 2, pp. 535-577 | Article | MR 3819702

[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 | Article | MR 1457054

[Kus67] Kushnirenko, A. G. Metric invariants of entropy type, Usp. Mat. Nauk, Volume 22 (1967) no. 5 (137), pp. 57-65 | MR 0217257

[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 | Article | MR 1709432

[Mar13] Marco, Jean-Pierre Polynomial entropies and integrable Hamiltonian systems, Regul. Chaotic Dyn., Volume 18 (2013) no. 6, pp. 623-655 | Article | MR 3146583 | Zbl 1286.70022

[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 1352607

[Nak95b] Nakayama, Hiromichi On dimensions of non-Hausdorff sets for plane homeomorphisms, J. Math. Soc. Japan, Volume 47 (1995) no. 4, pp. 789-793 | Article | MR 1348759

[Wal82] Walters, Peter An introduction to ergodic theory, Springer, Graduate Texts in Mathematics, Volume 79 (1982), ix+250 pages