Entropy along convex shapes, random tilings and shifts of finite type
type: Item
instance of: scholarly article
title: Entropy along convex shapes, random tilings and shifts of finite type
zbMATH Open document ID: 1025.37010
author: Béla Bollobás
author: Paul Balister
author: Anthony Quas
published in: Illinois Journal of Mathematics
publication date: 2003-11-25T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 37B50
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 37A35
zbMATH DE Number: 1865532
zbMATH Keywords: topological entropy
zbMATH Keywords: variational principle
zbMATH Keywords: \(d\)-dimensional subshift
zbMATH Keywords: symbolic dynamical system
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1865532