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 Σ_g on S~2 by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups MCG(S, (partial deriv)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.
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