One-sided almost specification and intrinsic ergodicity

Abstract

We define a new property called one-sided almost specification, which lies between the properties of specification and almost specification, and prove that it guarantees intrinsic ergodicity (i.e. uniqueness of the measure of maximal entropy) if the corresponding mistake function $g$ is bounded. We also show that uniqueness may fail for unbounded $g$ such as $\log \log n$. Our results have consequences for almost specification: we prove that almost specification with $g\equiv 1$ implies one-sided almost specification (with $g\equiv 1$) and hence uniqueness. On the other hand, the second author showed recently that almost specification with $g\equiv 4$ does not imply uniqueness.