Circularly ordered dynamical systems

Abstract

We study topological properties of circularly ordered dynamical systems and prove that every such system is representable on a Rosenthal Banach space, hence, is also tame. We derive some consequences for topological groups. We show that several Sturmian like symbolic Z k -systems are circularly ordered. Using some old results we characterize circularly ordered minimal cascades.

Group actions on treelike compact spaces

“…For motivation, properties and examples see, for example, [14,15,18] and the references thereof. A relevant recent result is that, for every circularly (in particular, linearly) ordered compact space K, the action H + (K) K, of the topological group H + (K) of all order-preserving homeomorphisms of K on K, is Rosenthal representable, [17].…”

Orderable groups and semigroup compactifications

“…Every (c-)ordered compact G-space K is tame in the sense of A. Köhler [24] (regular, in the original terminology). If K is metrizable then it is equivalent to say that the enveloping semigroup E(K) is "small"; namely, a separable Rosenthal compact space (see [15,16]). In view of a hierarchy of tame metric dynamical systems (see [19]) induced by the Todorcević' Trichotomy for Rosenthal compact spaces, we ask the following Question 1.4.…”

“…Ordered dynamical systems were studied in several recent publications concerning tame systems, Sturmian like circularly ordered symbolic systems and some dynamical generalizations of the amenability concept. See joint works with Eli Glasner [16,17,18,19] and also [27]. Investigation of order-preserving compact dynamical systems provides natural framework to study orderable groups (and the orderability itself).…”

“…The following natural topological generalization of (left) linear and circular orderability of abstract groups first was defined in [16]. Definition 1.1.…”

“…The second reason comes from our papers [16,17,19] with Eli Glasner, where we study Sturmian like and other circularly ordered dynamical systems. In many such cases, the enveloping semigroup is a circularly ordered semigroup.…”