Entropy zero area preserving diffeomorphisms of $S^2$

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.