Shifts of Finite Type with Nearly Full Entropy

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 δd > 0 such that for any nearest neighbor Z d shift of finite type X with alphabet A for which (log |A|) − h(X) < δd, X has a unique measure of maximal entropy µ. We also show that any such X is a measure-theoretic universal model in the sense of [25], that h(X) is a computable number, that µ is measure-theoretically isomorphic to a Bernoulli measure, and that the support of µ has topologically completely positive entropy. Though there are other sufficient conditions in the literature (see [9], [15], [22]) 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.