Extreme Relations for Topological Flows

Abstract

We introduce the concept of an extreme relation for a topological flow as an analogue of the extreme measurable partition for a measure-preserving transformation considered by Rokhlin and Sinai, and we show that every topological flow has such a relation for any invariant measure. From this result, it follows, among other things, that any deterministic flow has zero topological entropy and any flow which is a K-system with respect to an invariant measure with full support is a topological K-flow.

Local entropy theory

“…Building on results from [10] and [31] and from their earlier work [69] Kaminski, Seimaszko and Szymanski, in [70], tie up local entropy theory with classical results of Rohlin and Sinai [87]. Definition 13.6.…”

“…Part 4 is dedicated to further applications of the local theory. In Section 13 we discuss Li-Yorke chaos after Blanchard-Glasner-Kolyada-Maass [12], the relation of asymptotic pairs after Blanchard-Host-Ruette [10], and the notions of µ-Pinsker factor and extreme relations according to Kaminski-Siemaszko-Szymanski [70]. A new application to interval maps is proved in Theorem 13.9.…”

On the topological Kolmogorov property of the Chacon and Petersen subshifts

“…It has been shown in [4] that for any topological dynamical system (X, T ) and an invariant probability measure μ there exists an invariant and generating relation R ∈ CER(X) with…”

On Extensions over Semigroups and Applications

“…Moreover, Kamiński et al showed that a system (X, T) is topologically predictable if and only if every factor of (X, T) is invertible, where a factor is a system (Y, S) and a continuous onto map π : X → Y such that π • T = S • π. For Z-actions, it was shown in [2] that TP systems have zero topological entropy. Then, a natural question is whether this result also holds for general group actions with some natural modification of the definition of TP.…”