Dynamical compactness and sensitivity

Abstract

Abstract. To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system (X, T ) given by a compact metric space X and a continuous surjective self-map T : X → X. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

Auslander-Yorke type dichotomy theorems for stronger versions of $r$-sensitivity

“…The notion of sensitivity was generalized by measuring the set of nonnegative integers for which the sensitivity occurs [12,13,17,19,21,29]. For a subset A of natural numbers N, we say A is (1) thick if for any k ∈ N we can find some n ∈ N such that {n, n+1, · · · , n+k} ⊂ A; (2) syndetic if there exists some k ∈ N such that for every n ∈ N we have {n, n + 1, · · · , n + k} ∩ A = ∅; (3) thickly syndetic if {n ∈ Z + : {n, n + 1, · · · , n + k} ⊂ A} is syndetic for each k ∈ N. Thick sensitivity, thickly syndetical sensitivity and multi sensitivity were introduced and investigated in [21,20].…”

Finite Intersection Property and Dynamical Compactness

“…In [22] the authors consider one of possible dynamical compactness -transitive compactness, and its relations with well-known chaotic properties of dynamical systems. Let N T be the set of all subsets of Z + containing some N T (U, V ), where U, V are opene subsets of X.…”

“…Any transitive compact system is obviously topologically transitive, and observe that each weakly mixing 1 Remark that the notation ω F (x) used here is different from the one used in [1] (the notation ω F (x) used here is in fact ω kF (x) introduced in [1]). As this paper is a continuation of the research in [22], in order to avoid any confusion of notation or concept, we will follow the ones used in [22]. 2 A point x ∈ X is called recurrent if x ∈ ω T (x).…”

“…As a byproduct, we show that the ω NT -limit sets and the ω-limit sets have quite different topological structures for a general dynamical compact system (X, T ). At the end of this section we add one more chaotical property of transitive compact systems (in additional to already known from [22]): transitive compactness implies Li-Yorke chaos (Proposition 6.6).…”

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Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22]. In this paper we continue to investigate this notion.

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