One-sided almost specification and intrinsic ergodicity

TL;DR

This paper introduces the concept of one-sided almost specification, demonstrating that bounded mistake functions ensure unique maximal entropy measures in shift spaces, while unbounded functions can lead to non-uniqueness, refining understanding of ergodic properties.

Contribution

It defines one-sided almost specification and proves it guarantees intrinsic ergodicity for bounded mistake functions, clarifying conditions for uniqueness in shift spaces.

Findings

Bounded mistake function g guarantees unique measure of maximal entropy.

Unbounded g, such as log log n, can lead to multiple measures.

Almost specification with g=1 implies one-sided almost specification and uniqueness.

Abstract

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 log n. 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.