Shifts of finite type with nearly full entropy

TL;DR

This paper investigates Z^d shifts of finite type with entropy close to the maximum, proving they have unique measures of maximal entropy and other desirable properties, with bounds depending on the dimension.

Contribution

It establishes a new condition based on entropy proximity that guarantees uniqueness of the measure of maximal entropy without detailed adjacency rules.

Findings

Existence of a dimension-dependent threshold beta_d for entropy proximity.

Unique measure of maximal entropy for shifts with entropy within beta_d of maximum.

The measure of maximal entropy is Bernoulli and the entropy is computable.

Abstract

For any fixed alphabet A, the maximum topological entropy of a Z^d subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor Z^d shifts of finite type which have topological entropy very close to this maximum, and show that they have many useful properties. Specifically, we prove that for any d, there exists beta_d such that for any nearest neighbor Z^d shift of finite type X with alphabet A for which log |A| - h(X) < beta_d, X has a unique measure of maximal entropy. Our values of beta_d decay polynomially (like O(d^(-17))), and we prove that the sequence must decay at least polynomially (like d^(-0.25+o(1))). We also show some other desirable properties for such X, for instance that the topological entropy of X is computable and that the unique m.m.e. is isomorphic to a Bernoulli measure. Though there are other sufficient conditions in the literature which guarantee a unique measure of maximal entropy for Z^d shifts of finite type, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.