Seminar 03.11.21 Kwietniak

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

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

Zoom link:

Meeting ID: 980 3359 0349

Password: Mixing

Recorded talk: