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January 2020, 40(1): 319-329. doi: 10.3934/dcds.2020012
A shift map with a discontinuous entropy function
Christian Wolf ,
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).
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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.
Keywords: Rotation set, localized entropy, entropy spectrum, multifractal analysis, discontinuity.
Mathematics Subject Classification: Primary: 37A35, 37E45; Secondary: 37C40.
Citation: Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
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Figure 1. The set $ {\mathcal R} = {\mathcal R}(\Phi) $ in Example 1
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