One-sided Almost Specification and Intrinsic Ergodicity

Shift spaces with the specification property are intrinsically ergodic, i.e. they have a unique measure of maximal entropy. This can fail for shifts with the weaker almost specification property. We define a new property called one-sided almost specification, which lies in between specification and almost specification, and prove that it guarantees intrinsic ergodicity if the corresponding mistake function g is bounded. We also show that uniqueness may fail for unbounded g such as log logn. Our results have consequences for almost specification: we prove that almost specification with g = 1 implies one-sided almost specification (with g = 1), and hence uniqueness. On the other hand, the second author showed recently that almost specification with g = 4 does not imply uniqueness. This leaves open the question of whether almost specification implies intrinsic ergodicity when g = 2 or g = 3.