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