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Fall 2002 Entropy along convex shapes, random tilings and shifts of finite type
Paul Balister, Béla Bollobás, Anthony Quas
Illinois J. Math. 46(3): 781-795 (Fall 2002). DOI: 10.1215/ijm/1258130984


A well-known formula for the topological entropy of a symbolic system is $h_{\operatorname{top}}(X)=\lim_{n\to\infty} \log N(\Lambda_n)/|\Lambda_n|$, where $\Lambda_n$ is the box of side $n$ in $\mathbb{Z}^d$ and $N(\Lambda)$ is the number of configurations of the system on the finite subset $\Lambda$ of $\mathbb{Z}^d$. We investigate the convergence of the above limit for sequences of regions other than $\Lambda_n$ and show in particular that if $\Xi_n$ is any sequence of finite `convex' sets in $\mathbb{Z}^d$ whose inradii tend to infinity, then the sequence $\log N(\Xi_n)/|\Xi_n|$ converges to $h_{\operatorname{top}}(X)$. We apply this to give a concrete proof of a `strong Variational Principle', that is, the result that for certain higher dimensional systems the topological entropy of the system is the supremum of the measure-theoretic entropies taken over the set of all invariant measures with the Bernoulli property.


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Paul Balister. Béla Bollobás. Anthony Quas. "Entropy along convex shapes, random tilings and shifts of finite type." Illinois J. Math. 46 (3) 781 - 795, Fall 2002.


Published: Fall 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1025.37010
MathSciNet: MR1951240
Digital Object Identifier: 10.1215/ijm/1258130984

Primary: 37B50
Secondary: 37A35, 37B10, 37B40, 52C07

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign


Vol.46 • No. 3 • Fall 2002
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