One-sided almost specification and intrinsic ergodicity
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.
Keywords
Cite
@article{arxiv.1605.05354,
title = {One-sided almost specification and intrinsic ergodicity},
author = {Vaughn Climenhaga and Ronnie Pavlov},
journal= {arXiv preprint arXiv:1605.05354},
year = {2017}
}
Comments
27 pages. In v2, Theorem 3.5 is strengthened using arXiv:1702.02870, leading to a simpler proof of Theorem 3.8. Theorem 3.1 has been reformulated in terms of the measure center