  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.
