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

  • ANTHONY QUAS (a1) and AYŞE A. ŞAHİN (a2)

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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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