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. As an 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 $MCG(S, \partial S)$ with non-trivial first cohomology.