  For a continuous map $f$ on a compact metric space we study the geometry and
entropy of the generalized rotation set $\R(\Phi)$. Here
$\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and
$\R(\Phi)$ is the set of all $\mu$-integrals of $\Phi$ and $\mu$ runs over all
$f$-invariant probability measures. It is easy to see that the rotation set is
a compact and convex subset of $\bR^m$. We study the question if every compact
and convex set is attained as a rotation set of a particular set of potentials
within a particular class of dynamical systems. We give a positive answer in
the case of subshifts of finite type by constructing for every compact and
convex set $K$ in $\bR^m$ a potential $\Phi=\Phi(K)$ with $\R(\Phi)=K$. Next,
we study the relation between $\R(\Phi)$ and the set of all statistical limits
$\R_{Pt}(\Phi)$. We show that in general these sets differ but also provide
criteria that guarantee $\R(\Phi)= \R_{Pt}(\Phi)$. Finally, we study the
entropy function $w\mapsto H(w), w\in \R(\Phi)$. We establish a variational
principle for the entropy function and show that for certain non-uniformly
hyperbolic systems $H(w)$ is determined by the growth rate of those hyperbolic
periodic orbits whose $\Phi$-integrals are close to $w$. We also show that for
systems with strong thermodynamic properties (subshifts of finite type,
hyperbolic systems and expansive homeomorphisms with specification, etc.) the
entropy function $w\mapsto H(w)$ is real-analytic in the interior of the
rotation set.
