Ergodic Theory and Dynamical Systems



Entropy gaps and locally maximal entropy in $\mathbb{Z}^d$ subshifts


ANTHONY QUAS a1 and AYSE A. SAHIN a2
a1 Department of Mathematical Sciences, Dunn Hall, University of Memphis, Memphis, TN 38152-3240, USA (e-mail: quasa@msci.memphis.edu)
a2 Department of Mathematics, DePaul University, 2320 North Kenmore St, Chicago, IL 60614, USA (e-mail: asahin@condor.depaul.edu)

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Abstract

In this paper, we study the behaviour of the entropy function of higher-dimensional shifts of finite type. We construct a topologically mixing $\mathbb{Z}^2$ shift of finite type whose ergodic invariant measures are connected in the $\overline{d}$ topology and whose entropy function has a strictly local maximum. We also construct a topologically mixing $\mathbb{Z}^2$ shift of finite type X with the property that there is a uniform gap between the topological entropy of X and the topological entropy of any subshift of X with stronger mixing properties. Our examples illustrate the necessity of strong topological mixing hypotheses in existing higher-dimensional representation and embedding theorems.

(Received December 8 2001)
(Accepted in revised form May 24 2002)