Shifts of finite type with nearly full entropy
Ronnie Pavlov · Proceedings of the London Mathematical Society · 2013
For any fixed alphabet A, the maximum topological entropy of a ℤd subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor ℤd shifts of finite type (SFTs) 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 βd such that, for any nearest neighbor ℤd SFT X with alphabet A for which (log |A|)−h(X)<βd, X has a unique measure of maximal entropy μ. Our values of βd 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 μ is isomorphic to a Bernoulli measure. Although there are other sufficient conditions in the literature (see [Burton and Steif, Israel J. Math. 89 (1995) 275–300; Häggström, Israel J. Math. 94 (1996) 319–352; Markley and Paul, Lect. Notes Pure Appl. Math. 70 (1981) 135–157]) which guarantee a unique measure of maximal entropy for ℤd SFTs, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.