  Equilibrium states are natural dynamical analogs of Gibbs states in
thermodynamic formalism. This paper is devoted to the study of their
computability in the sense of Computable Analysis. We show that the unique
equilibrium state associated to a pair of a computable, topologically exact,
distance-expanding, open map $T\colon X\rightarrow X$ and a computable H\"older
continuous potential $\varphi\colon X\rightarrow\mathbb{R}$ is always
computable. Furthermore, the Hausdorff dimension of the Julia set and the
equilibrium state for the geometric potential of a computable hyperbolic
rational map are computable. On the other hand, we introduce a mechanism to
establish non-uniqueness of equilibrium states. We also present some computable
dynamical systems whose equilibrium states are all non-computable.
