  We study the interplay between the dynamics of area-preserving surface
homeomorphisms homotopic to the identity and the topology of the surface. We
define fully essential dynamics and generalize the results previously obtained
on strictly toral dynamics to surfaces of higher genus. Non-fully essential
dynamics are, in a way, reducible to surfaces of lower genus, while in the
fully essential case the dynamics is decomposed into a disjoint union of
periodic bounded disks and a complementary invariant externally transitive
continuum $C$. When the Misiurewicz-Ziemian rotation set has non-empty interior
the dynamics is fully essential, and the set $C$ is (externally) sensitive on
initial conditions and realizes all the rotational dynamics. As a fundamental
tool we introduce the notion of homotopically bounded sets and we prove a
general boundedness result for invariant open sets when the fixed point set is
inessential.
