Entropy zero area preserving diffeomorphisms of $S^2$

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 ${\mathrm{\Sigma }}_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG(S,\mathrm{\partial }S)$ with non-trivial first cohomology.