Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms

Militon, Emmanuel

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 565-604.

Soit $S$ une surface compacte connexe et soit $f$ un élément du groupe ${\mathrm{Homeo}}_0\left(S\right)$ des homéomorphismes de $S$ isotopes à l’identité. Notons $\tilde{f}$ un relevé de $f$ au revêtement universel de $S$. Fixons un domaine fondamental $D$ de ce revêtement universel. On dit que l’homéomorphisme $f$ est non-diffus si la suite $(d_n/n)$ converge vers $0$, où $d_n$ désigne le diamètre de ${\tilde{f}}^n\left(D\right)$. Supposons que la surface $S$ est orientable de bord non vide. Nous démontrons que, si $S$ est distincte du disque et de l’anneau, un homéomorphisme est non-diffus si et seulement si il a des conjugués dans ${\mathrm{Homeo}}_0\left(S\right)$ arbitrairement proches de l’identité. Dans le cas où la surface $S$ est l’anneau, nous démontrons qu’un homéomorphisme est non-diffus si et seulement si il a des conjugués arbitrairement proches d’une rotation (ce résultat était déjà connu dans la plupart des cas grâce à un théorème dû à Béguin, Crovisier, Le Roux et Patou). On en déduit que, pour de telles surfaces $S$, un élément de ${\mathrm{Homeo}}_0\left(S\right)$ est distordu si et seulement si il est non diffus.

Let $S$ be a compact connected surface and let $f$ be an element of the group ${\mathrm{Homeo}}_0\left(S\right)$ of homeomorphisms of $S$ isotopic to the identity. Denote by $\tilde{f}$ a lift of $f$ to the universal cover of $S$. Fix a fundamental domain $D$ of this universal cover. The homeomorphism $f$ is said to be non-spreading if the sequence $(d_n/n)$ converges to $0$, where $d_n$ is the diameter of ${\tilde{f}}^n\left(D\right)$. Let us suppose now that the surface $S$ is orientable with a nonempty boundary. We prove that, if $S$ is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in ${\mathrm{Homeo}}_0\left(S\right)$ arbitrarily close to the identity. In the case where the surface $S$ is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in ${\mathrm{Homeo}}_0\left(S\right)$ arbitrarily close to a rotation (this was already known in most cases by a theorem by Béguin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces $S$, an element of ${\mathrm{Homeo}}_0\left(S\right)$ is distorted if and only if it is non-spreading.

Classification : 54H20, 54H15, 37C85
Keywords: Topological dynamics, homeomorphism, distortion, conjugacy class
Mot clés : Dynamique topologique, homéomorphisme, distorsion, classe de conjugaison

[1] Béguin, F.; Crovisier, S.; Le Roux, F.; Patou, A. Pseudo-rotations of the closed annulus: variation on a theorem of J. Kwapisz, Nonlinearity, Volume 17 (2004) no. 4, pp. 1427-1453 | DOI | MR | Zbl

[2] Bounemoura, A. Simplicité des groupes de transformations de surfaces, Ensaios Matemáticos, 14, Sociedade Brasileira de Matemática, Rio de Janeiro, 2008 | MR | Zbl

[3] Calegari, Da.; Freedman, M. H. Distortion in transformation groups, Geom. Topol., Volume 10 (2006), pp. 267-293 (with an appendix by Yves de Cornulier) | DOI | MR

[4] Epstein, D. B. A. Curves on $2$-manifolds and isotopies, Acta Math., Volume 115 (1966), pp. 83-107 | DOI | MR | Zbl

[5] Farb, B.; Margalit, D. A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012 | MR

[6] Fisher, G. M. On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc., Volume 97 (1960), pp. 193-212 | DOI | MR | Zbl

[7] Franks, J.; Handel, M. Distortion elements in group actions on surfaces, Duke Math. J., Volume 131 (2006) no. 3, pp. 441-468 | DOI | MR | Zbl

[8] Hamstrom, M.-E. Homotopy groups of the space of homeomorphisms on a $2$-manifold, Illinois J. Math., Volume 10 (1966), pp. 563-573 | MR

[9] de la Harpe, P. Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000 | Zbl

[10] Katok, A.; Hasselblatt, B. Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995 (with a supplementary chapter by Katok and Leonardo Mendoza) | MR | Zbl

[11] Kwapisz, J. Combinatorics of torus diffeomorphisms, Ergodic Theory Dynam. Systems, Volume 23 (2003) no. 2, pp. 559-586 | DOI | MR | Zbl

[12] Le Calvez, P.; Yoccoz, J.-C. Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe, Ann. of Math. (2), Volume 146 (1997) no. 2, pp. 241-293 | DOI | Zbl

[13] Militon, E. Distortion elements for surface homeomorphisms, Geom. Topol., Volume 18 (2014) no. 1, pp. 521-614 | DOI | MR | Zbl

[14] Misiurewicz, M.; Ziemian, K. Rotation sets for maps of tori, J. London Math. Soc. (2), Volume 40 (1989) no. 3, pp. 490-506 | DOI | MR

[15] Polterovich, L. Growth of maps, distortion in groups and symplectic geometry, Invent. Math., Volume 150 (2002) no. 3, pp. 655-686 | DOI | MR | Zbl