A shift map with a discontinuous entropy function

Let $f:X\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $μ\mapsto h_μ(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $φ:X\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $Φ:X\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $Φ$ with a discontinuous localized entropy function.