Entropy zero area preserving diffeomorphisms of S2

Franks, John¹ ; Handel, Michael²

¹ Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States
² Mathematics & Computer Science Department, Herbert H Lehman College, CUNY, Bronx, NY 10468-1589, United States

Geometry & topology, Tome 16 (2012) no. 4, pp. 2187-2284.

Résumé

In this paper we formulate and prove a structure theorem for area preserving diffeomorphisms of genus zero surfaces with zero entropy and at least three periodic points. As one application we relate the existence of faithful actions of a finite index subgroup of the mapping class group of a closed surface ${\Sigma }_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG\left(S,\partial S\right)$ with nontrivial first cohomology. In another application we show that the rotation number is defined and continuous at every point of a zero entropy area preserving diffeomorphism of the annulus.

Détail

DOI : 10.2140/gt.2012.16.2187

Classification : 37C05, 37C85
Keywords: entropy zero diffeomorphism

Bibliographie

[1] R Bowen, Entropy and the fundamental group, from: "The structure of attractors in dynamical systems" (editors N G Markley, J C Martin, W Perrizo), Lecture Notes in Math. 668, Springer (1978) 21

[2] M Brown, J M Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984) 503

[3] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge Univ. Press (1988)

[4] B Farb, Some problems on mapping class groups and moduli space, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 11

[5] A Fathi, F Laudenbach, V Poenaru, Editors, Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France (1979) 284

[6] D Fisher, Groups acting on manifolds: around the Zimmer program, from: "Geometry, rigidity, and group actions" (editors B Farb, D Fisher), Univ. Chicago Press (2011) 72

[7] J Franks, Generalizations of the Poincaré–Birkhoff theorem, Ann. of Math. 128 (1988) 139

[8] J Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8∗ (1988) 99

[9] J Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math. 2 (1996) 1

[10] J Franks, Distortion in groups of circle and surface diffeomorphisms, from: "Dynamique des difféomorphismes conservatifs des surfaces: un point de vue topologique", Panor. Synthèses 21, Soc. Math. France (2006) 35

[11] J Franks, M Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol. 7 (2003) 713

[12] J Franks, M Handel, Distortion elements in group actions on surfaces, Duke Math. J. 131 (2006) 441

[13] J Franks, M Handel, Global fixed points for centralizers and Morita's theorem, Geom. Topol. 13 (2009) 87

[14] M Handel, A pathological area preserving $C^{\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982) 163

[15] M Handel, The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phys. 127 (1990) 339

[16] M Handel, Commuting homeomorphisms of $S^2$, Topology 31 (1992) 293

[17] M Handel, A fixed-point theorem for planar homeomorphisms, Topology 38 (1999) 235

[18] M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1994)

[19] N V Ivanov, Fifteen problems about the mapping class groups, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 71

[20] A Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. (1980) 137

[21] R Kirby, Problems in low dimensional manifold theory, from: "Algebraic and geometric topology. Part 2." (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 273

[22] M Korkmaz, Problems on homomorphisms of mapping class groups, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 81

[23] J N Mather, Topological proofs of some purely topological consequences of Carathéodory's theory of prime ends, from: "Selected studies: physics-astrophysics, mathematics, history of science" (editors T M Rassias, G M Rassias), North-Holland (1982) 225

[24] J D Mccarthy, On the first cohomology group of cofinite subgroups in surface mapping class groups, Topology 40 (2001) 401

[25] L Polterovich, Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150 (2002) 655

[26] C P Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math. 26 (1974) 187

[27] W P Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974) 347

[28] Y Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987) 285