Dynamical Resistance Day

Dbar-approachability, entropy density and B-free shifts
18 de Dezembro | 12:10 | Dominik Kwietniak (Jagiellonian Univ. in Krakow, Poland)

Title: Dbar-approachability, entropy density and B-free shifts (video)

We study which properties of shift spaces (shifts) are closed with respect to Hausdorff metric dbar (H-dbar) limits. In particular, we study shifts, which are H-dbar limits of their Markov approximations. We call these shifts dbar-approachable. We provide a topological
characterisation of chain mixing dbar-approachable shifts analogous to Friedman-Ornstein's characterisation of Bernoulli processes.

We prove that many specification properties imply dbar-approachability. It follows that mixing shifts of finite type, mixing sofic shifts, beta-shifts are dbar-approachable . We construct minimal and proximal examples of mixing dbar-approachable shifts. We also show that dbar-approachability and chain-mixing imply dbar-stability, a property recently introduced by Tim Austin. This allows us to provide examples of minimal or proximal dbar-stable shift spaces, answering a question posed by Austin. Finally, we show that the set of shifts with entropy-dense ergodic measures is H-dbar closed. Note that entropy-density of ergodic measures is known to hold for shifts with the specification property but our technique yields entropy-density for examples which are far from having any specification. Finally, we apply our technique to a class of shifts including many interesting B-free shifts. These shift spaces are not dbar-approachable, but they
are easily seen to be approximated by naturally defined sequences of transitive sofic shifts, and this implies entropy-density.

This is a joint work with Jakub Konieczny and Michal Kupsa.