Search for: All records

Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ .