Dynamical compactness and sensitivity

TL;DR

This paper introduces the concept of dynamically compact systems in chaos theory, linking topological transitivity with various sensitivity properties, and explores their implications and relationships within dynamical systems.

Contribution

It defines dynamically compact systems and establishes their connections with weak mixing, sensitivity, and entropy, providing new insights into chaotic dynamics.

Findings

Weakly mixing systems are transitive compact.

Transitive compact M-systems are weakly mixing.

Multi-sensitivity implies positive topological sequence entropy.

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 $T:X \to X$. 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.