Entropy along convex shapes, random tilings and shifts of finite type
Let \(X\) be a symbolic dynamical system. If \(\Lambda_{n}\) denotes 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 \(\mathbb{Z}\), then \(\lim_{n\rightarrow \infty}\log N(\Lambda_{n})/|\Lambda_{n}|\) exists and it is called the topological entropy of \(X\) which is denoted by \(h_{\text{top}}(X)\). Clearly, the quantity \(N(\Lambda)\) can be defined for any finite subset \(\Lambda\) of \(\mathbb{Z}^{d} \). So, a natural question arises: can one consider more general sequences \(\Xi_{n}\) of regions than \(\Lambda_{n}\) with the property that \(\log N(\Xi_{n})/|\Xi_{n}|\) converges to \(h_{\text{top}}(X)\)? The paper under review deals with this question. The authors prove that \(\Xi_{n}\) can be chosen as any sequence of finite convex sets in \(\mathbb{Z}^{d}\) whose inradii tend to infinity and they apply this result to give a proof of a variational principle. The results of the paper are based in the following two main theorems. Recall that if \(d\geq 1\) and \({\mathcal F}\) denotes the collection of finite subsets of \(\mathbb{Z}^{d}\), a functional \(\alpha : {\mathcal F}\rightarrow \mathbb{R}\) is called monotonic if \(\Lambda\subset \Lambda'\) implies \(\alpha (\Lambda)\leq \alpha (\Lambda')\) and subadditive if for disjoint subsets \(\Lambda\) and \(\Lambda'\), \(\alpha(\Lambda\cup\Lambda')\leq \alpha(\Lambda)+\alpha (\Lambda')\). If \(\alpha (\Lambda+v)=\alpha (\Lambda)\) for all \(v\in {\mathcal F}\), then \(\alpha\) is said to be translation-invariant. Theorem 1. Let \(\alpha :{\mathcal F}\rightarrow \mathbb{R}\) be a monotonic translation-invariant subadditive functional. Given \(\varepsilon >0\), there exists \(r>0\) such that if \(C\) is any bounded convex set in \(\mathbb{R}^{d}\) containing a ball of radius \(r\), then the distance from \( \alpha (C\cap \mathbb{Z}^{d})/|C\cap \mathbb{Z}^{d}|\) to \(\inf \{\alpha (\Lambda_{n})/|\Lambda_{n}|: n\in \mathbb{N}\}\) is less or equal to \(\varepsilon\). Theorem 2. Let \(X\) be a strongly irreducible shift of finite type. Then, for all \(\varepsilon >0\), there is a Bernoulli measure \(\mu\) on \(X\) such that \(h_{\mu}(X)\geq h_{\text{top}}(X)-\varepsilon\), where \(h_{\mu}(X)\) is the measure-theoretic entropy of \(X\) with respect to \(\mu\).
scientific article; zbMATH DE number 1865532