Entropy zero area preserving diffeomorphisms of S²

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 ∑_g 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.