#### Volume 16, issue 4 (2012)

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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 ${\Sigma }_{g}$ 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.

##### Keywords
entropy zero diffeomorphism
##### Mathematical Subject Classification 2010
Primary: 37C05, 37C85