A shift map with a discontinuous entropy function

Christian Wolf

doi:10.3934/dcds.2020012

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2020, Volume 40, Issue 1: 319-329. Doi: 10.3934/dcds.2020012

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$ L^{2} $

initial data Next Article Fermi's golden rule and

$ H^1 $

scattering for nonlinear Klein-Gordon equations with metastable states

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
^* Corresponding author: Christian Wolf

Received: December 31, 2018

Revised: June 30, 2019

Published: October 2019

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

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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 {\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.

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Mathematics Subject Classification: Primary: 37A35, 37E45; Secondary: 37C40.

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Figure 1. The set $ {\mathcal R} = {\mathcal R}(\Phi) $ in Example 1

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