# American Institute of Mathematical Sciences

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January  2020, 40(1): 319-329. doi: 10.3934/dcds.2020012

## A shift map with a discontinuous entropy function

 Department of Mathematics, The City College of New York, New York, NY, 10031, USA

* Corresponding author: Christian Wolf

Received  January 2019 Revised  July 2019 Published  October 2019

Fund Project: Christian Wolf was partially supported by a grant from the PSC-CUNY (TRADB-49-253 to Christian Wolf)

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 $\phi:X\to {\mathbb R}$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\Phi:X\to {\mathbb R}^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\Phi$ with a discontinuous localized entropy function.

Citation: Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
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##### References:
The set ${\mathcal R} = {\mathcal R}(\Phi)$ in Example 1