Dbar-approachability, entropy density and B-free shifts

Abstract: We study which properties of shift spaces transfer to their Hausdorff metric dbar-limits. In particular, we study shift spaces we call dbar-approachable, which are Hausdorff metric dbar-limits of their own k-step Markov approximations. We provide a topological characterisation of chain mixing dbar-approachable shift spaces using the dbar-shadowing property. This can be considered as an analogue for Friedman and Ornstein's characterisation of Bernoulli processes. We prove that many classical specification properties imply chain mixing and dbar-approachability. It follows that there are tons of interesting dbar-approachable shift spaces (mixing shifts of finite type, or more generally mixing sofic shifts, or even more generally, shift spaces with the specification or beta-shifts. In addition, we construct minimal and proximal examples of dbar-approachable shift spaces, thus proving dbar-approachability is a more general phenomenon than specification. We also show that dbar-approachability and chain-mixing imply dbar-stability, a property recently introduced by Tim Austin in his study of Bernoulliness of equilibrium states. This allows us to provide first examples of minimal or proximal dbar-stable shift spaces, thus answering a question posed by Austin. Finally, we show that the set of shift spaces with entropy-dense ergodic measures is closed wrt dbar Hausdorff metric. Note that entropy-density of ergodic measures is known to hold for many classes of shift spaces with variants of the specification property, but our result show that in these cases the entropy-density is a mere consequence of entropy-density of mixing shifts of finite type and dbar-approachability. Since we know there are examples of minimal or proximal dbar-approachable shifts, we see that our technique yields entropy-density for examples which were beyond the reach of methods based on specification properties. Finally, we apply our technique to hereditary closures of B-free shifts (a class including many interesting B-free shifts). These shift spaces are not chain-mixing, hence they 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.

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University