  Given a continuous dynamical system $f:X\to X$ on a compact metric space $X$
and an $m$-dimensional continuous potential $\Phi:X\to \mathbb R^m$, the
(generalized) rotation set ${\rm Rot}(\Phi)$ is defined as the set of all
$\mu$-integrals of $\Phi$, where $\mu$ runs over all invariant probability
measures. Analogous to the classical topological entropy, one can associate the
localized entropy ${\mathcal H}(w)$ to each $w\in {\rm Rot}(\Phi)$. In this
paper, we study the computability of rotation sets and localized entropy
functions by deriving conditions that imply their computability. We then apply
our results to study to the case of subshifts of finite type. We prove that
${\rm Rot}(\Phi)$ is computable and that ${\mathcal H}(w)$ is computable in the
interior of the rotation set. Finally, we construct an explicit example that
shows that, in general, ${\mathcal H}$ is not continuous on the boundary of the
rotation set, when considered as a function of $\Phi$ and $w$. This suggests
that, in general, ${\mathcal H}$ is not computable at the boundary of rotation
sets.
