**Title:** Dbar-approachability, entropy density and B-free shifts

**Speaker:** Dominik Kwietniak – Jagiellonian University in Krakow

**Abstract:** Let dbar denote the pseudometric on the full shift over a

finite alphabet A given by the upper asymptotic density of the set of

positions at which two A-valued sequences differ. Write H-dbar for the

associated Hausdorff pseudometric for subsets of the full shift. We

study which properties of shift spaces (shifts) are closed with

respect to H-dbar. In particular, we study shifts, which are H-dbar

limits of their Markov approximations. We call these shifts

dbar-approachable. We provide a topological characterization of chain

mixing dbar-approachable shifts analogous to Friedman-Ornstein’s

characterization of Bernoulli processes.

We prove that many specification properties imply

dbar-approachability. It follows that mixing shifts of finite type,

mixing sofic shifts, and 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

leads to 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 follow from the

specification property, but the minimal or proximal examples are far

from having any specification. Finally, we show entropy-density for a

class of shifts that includes many interesting B-free shifts. These

shift spaces are not dbar-approachable, but they are H-dbar limits of

sequences of transitive sofic shifts, and this implies

entropy-density.

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

**Zoom link:** https://osu.zoom.us/j/98033590349

**Meeting ID:** 980 3359 0349

**Password:** Mixing

**Recorded talk:** https://osu.zoom.us/rec/play/_DXkoWtXTB92Pui6F7zl4eoVstNWH1rMUdb2a8NjFe61zd2BC9dTZP4UnuUKAC9behs6MQs88XEToF8A.vhVYz7t7fI5_en7U?continueMode=true&_x_zm_rtaid=HImgc_KTTByZM_8W1gjyuA.1615523637945.3021b693a8b3eeed460d5a4c44061f1c&_x_zm_rhtaid=457