On topological models of zero entropy loosely Bernoulli systems

Abstract

We provide a purely topological characterisation of uniquely ergodic topological dynamical systems (TDSs) whose unique invariant measure is zero entropy loosely Bernoulli (following Ratner, we call such measures loosely Kronecker). At the heart of our proofs lies Feldman-Katok continuity (FK-continuity for short), that is, continuity with respect to the change of metric to the Feldman-Katok pseudometric. Feldman-Katok pseudometric is a topological analog of f-bar (edit) metric for symbolic systems. We also study an opposite of FK-continuity, coined FK-sensitivity. We obtain a version of Auslander-Yorke dichotomies: minimal TDSs are either FK-continuous or FK-sensitive, and transitive TDSs are either almost FKcontinuous or FK-sensitive.

On Feldman-Katok metric and entropy formulae

“…First we give the definition of Feldman-Katok metric [7]: Let (X , d, T ) be a TDS. For x, y ∈ X , δ > 0 and n ∈ N, we define an (n, δ )-match of x and y to be an order preserving (i.e.…”

“…In [7], the authors used the notion of µ-F K-equicontinuity to characterize loosely Kronecker system. Here we can similarly define the notion of µ-F -equicontinuous: Definition 3.5.…”

“…In this section we give some notes on [7], which may help us to understand how Feldman-Katok metric works.…”

“…The Feldman-Katok metric was introduced in [15] as a topological version of the edit distance f . In [7], the authors used it to characterize zero entropy loosely Bernoulli(the authors called it loosely Kronecker) MPSs and their topological models.…”

“…In section 3, we study another pseudo-metric related to Feldman-Katok metric. In section 4, we give some notes on [7].…”