Implications of Mixing Properties on Intrinsic Ergodicity in Symbolic Dynamics

We study various mixing properties for subshifts, which allow words in the language to be concatenated into a new word in the language given certain gaps between them. All are defined in terms of an auxiliary gap function f : N → N, which gives the minimum required gap length as a function of the lengths of the words on either side. In this work we focus mostly on topological transitivity, topological mixing, and a property which we call non-uniform two-sided specification. It was shown in [12] that non-uniform specification with gap function Ω(lnn) does not imply uniqueness of the measure of maximal entropy. In this work, we show that for all other gap functions, i.e. for all f(n) with lim inf f(n) lnn = 0, even the weaker property of non-uniform two-sided specification implies uniqueness of the measure of maximal entropy, and its full support. We also show that even the very weak property of topological transitivity implies uniqueness of the measure of maximal entropy (and its full support) when lim f(n) lnn = 0. We then define a class of subshifts satisfying this hypothesis, each of which therefore has a unique measure of maximal entropy. We also give classes of examples which demonstrate two negative results. The first is that multiple measures of maximal entropy can occur for subshifts with topological mixing with gap function which grows arbitrarily slowly along a subsequence. The second is that topological mixing can occur with arbitrarily slowly growing unbounded gap function for subshifts without any periodic points.