Eventual nonsensitivity and tame dynamical systems

Abstract

In this paper we characterize tame dynamical systems and functions in terms of eventual non-sensitivity and eventual fragmentability. As a notable application we obtain a neat characterization of tame subshifts X ⊂ {0, 1} Z : for every infinite subset L ⊆ Z there exists an infinite subset K ⊆ L such that π K (X) is a countable subset of {0, 1} K . The notion of eventual fragmentability is one of the properties we encounter which indicate some "smallness" of a family. We investigate a "smallness hierarchy" for families of continuous functions on compact dynamical systems, and link the existence of a "small" family which separates points of a dynamical system (G, X) to the representability of X on "good" Banach spaces. For example, for metric dynamical systems the property of admitting a separating family which is eventually fragmented is equivalent to being tame. We give some sufficient conditions for coding functions to be tame and, among other applications, show that certain multidimensional analogues of Sturmian sequences are tame. We also show that linearly ordered dynamical systems are tame and discuss examples where universal dynamical systems associated with certain Polish groups are tame. Contents 23 8. Order preserving systems are tame 31 9. Intrinsically tame groups 38 10. Appendix 41 References 43

Group actions on treelike compact spaces

“…In [16,30] we proved that every linear order preserving function on a compact linearly ordered topological space is fragmented. The following theorem is a result in the same spirit.…”

“…Now recall that the "generalized arc" [u, v] is a linearly ordered compact connected space, [33]. By results of Nachbin [34, p. 48 and 113] we have an order-preserving (hence, monotone in the sense of Definition 3.4.1) continuous map h : Finally note that it is straightforward to see that Rosenthal representability of any compact dynamical H(K)-system K implies that the topological group H(K) is Rosenthal representable (for details see for example [16,Lemma 3.5]). Proof.…”

“…for all finite disjoint subsets P, M of N. As in [16,18] we say that a bounded family of real valued functions is a tame family if it does not contain an independent sequence. For compact metrizable systems X the two concepts discussed above coincide: (G, X) is tame iff it is Rosenthal representable.…”

Compact operator semigroups applied to dynamical systems

“…A metric topological dynamical system (K; S) is tame if for each f ∈ C(K) the orbit T S f is relatively sequentially compact with respect to the product topology of C K . Such systems have been studied in detail by E. Glasner and M. Megrelishvili ([GM06], [Gla06], [Gla07a], [Gla07b], [GM12], [GM13] and [GM15]) and later by W. Huang in [Hua06], D. Kerr and H. Li in [KL07] as well as by A. Romanov in [Rom16]. We recall that a topological space X is a Fréchet-Urysohn space (see page 53 of [Eng89]) if each subset A ⊆ X satisfies…”

Order and minimality of some topological groups

“…Note that the groups H + (X) play a major role in many research lines. See, for example, [24,33,25].…”