Abstract
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.
Citation
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. https://doi.org/10.1215/ijm/1258130984
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