In this paper we formulate and prove a structure theorem for area preserving
diffeomorphisms of $S^2$ 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.