A shift map with a discontinuous entropy function (Q2281572)

Let \(f : X \rightarrow X\) be a continuous map on a compact metric space \(X\) with finite topological entropy. If one assumes that the entropy map \(\mu \rightarrow h_\mu (f)\) is upper semi-continuous, then the localized entropy function of a given continuous potential \(\Phi : X \rightarrow {\mathbb R}\) is continuous. The author proves that the above result does not hold for the case of higher-dimensional potentials \(\Phi : X \rightarrow {\mathbb R}^n\), \(n\geq 2\). Precisely, he constructs for a shift map \(f\) a \(2\)-dimensional Lipschitz continuous potential \(\Phi\) with a discontinuous localized entropy function.