  We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms
$f:S\rightarrow S$ homotopic to the identity. A set of hypotheses is presented,
called fully essential system of curves $\mathscr{C}$ and it is shown that
under these hypotheses, the natural lift of $f$ to the universal cover of $S$
(the Poincar\'e disk $\mathbb{D}),$ denoted $\widetilde{f},$ has complicated
and rich dynamics. In this context we generalize results that hold for
homeomorphisms of the torus homotopic to the identity when their rotation sets
contain zero in the interior. In particular, we prove that if $f$ is a
$C^{1+\epsilon }$ diffeomorphism for some $\epsilon >0$ and $\pi
:\mathbb{D}\rightarrow S$ is the covering map, then there exists a contractible
hyperbolic $f$-periodic saddle point $p\in S$ such that for any
$\widetilde{p}\in \pi ^{-1}(p),$ $$W^u(\widetilde{p}) \pitchfork
W^s(g(\widetilde{p})) $$ for all deck transformations $g\in Deck(\pi ).$ By
$\pitchfork,$ we mean a topologically transverse intersection between the
manifolds, see the precise definition in subsection 1.1. We also show that the
homological rotation set of such a $f$ is a compact convex subset of
$\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are
realized by compact $f$-invariant sets, periodic orbits in the rational case,
and $f$ has uniformly bounded displacement with respect to rotation vectors in
the boundary of the rotation set. Something that implies, in case $f$ is
area-preserving, that the rotation vector of Lebesgue measure belongs to the
interior of the rotation set.
