  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.
