A shift map with a discontinuous entropy function

TL;DR

This paper demonstrates that the localized entropy function can be discontinuous for higher-dimensional potentials, even when the entropy map is upper semi-continuous, contrasting with the one-dimensional case.

Contribution

It constructs a specific example of a shift map with a Lipschitz continuous potential where the localized entropy function is discontinuous, showing a fundamental difference in higher dimensions.

Findings

Localized entropy function can be discontinuous in higher dimensions

Upper semi-continuity of the entropy map does not guarantee continuity of localized entropy

Constructs a concrete example with a 2D Lipschitz potential

Abstract

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 $\mu\mapsto h_\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $\phi:X\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\Phi:X\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\Phi$ with a discontinuous localized entropy function.