Entropy along convex shapes, random tilings and shifts of finite type

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.