  We consider a continuous dynamical system $f:X\to X$ on a compact metric
space $X$ equipped with an $m$-dimensional continuous potential
$\Phi=(\phi_1,\cdots,\phi_m):X\to \bR^m$. We study the set of ground states $
GS(\alpha)$ of the potential $\alpha\cdot \Phi$ as a function of the direction
vector $\alpha\in S^{m-1}$. %We also study the corresponding rotation vectors
$\rv(GS(\alpha))$. We show that the structure of the ground state sets is
naturally related to the geometry of the generalized rotation set of $\Phi$. In
particular, for each $\alpha$ the set of rotation vectors of $ GS(\alpha)$
forms a non-empty, compact and connected subset of a face $F_\alpha(\Phi)$ of
the rotation set associated with $\alpha$. Moreover, every ground state
maximizes entropy among all invariant measures with rotation vectors in
$F_\alpha(\Phi)$. We further establish the occurrence of several quite
unexpected phenomena. Namely, we construct for any $m\in\bN$ examples with an
exposed boundary point (i.e. $F_\alpha(\Phi)$ being a singleton) without a
unique ground state. Further, we establish the possibility of a line segment
face $F_\alpha(\Phi)$ with a unique but non-ergodic ground state. Finally, we
establish the possibility that the set of rotation vectors of $GS(\alpha)$ is a
non-trivial line segment.
