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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Entropy zero area preserving diffeomorphisms of $S^2$

John Franks and Michael Handel

Geometry & Topology 16 (2012) 2187–2284
Abstract

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 Σg on S2 by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups MCG(S,S) 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.

Keywords
entropy zero diffeomorphism
Mathematical Subject Classification 2010
Primary: 37C05, 37C85
References
Publication
Received: 7 September 2010
Revised: 30 January 2012
Accepted: 25 July 2012
Published: 16 January 2013
Proposed: Danny Calegari
Seconded: Dmitri Burago, Leonid Polterovich
Authors
John Franks
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
United States
Michael Handel
Mathematics & Computer Science Department
Herbert H Lehman College
CUNY
Bronx, NY 10468-1589
United States