Finite intersection property and dynamical compactness

TL;DR

This paper explores the concept of dynamical compactness relative to families, linking it to the finite intersection property, and investigates its implications for weak mixing, disjointness, and sensitivity in dynamical systems, including linear dynamics.

Contribution

It establishes the equivalence between dynamical compactness and the finite intersection property for Furstenberg families and explores its implications for various dynamical behaviors.

Findings

Dynamical systems are compact with respect to a Furstenberg family iff the family has the finite intersection property.

Differences between $oldsymbol{ ext{omega}}_{oldsymbol{ ext{F}}}$-limit and $oldsymbol{ ext{omega}}$-limit sets are highlighted.

Equivalence of multi-sensitivity, sensitive compactness, and transitive sensitivity in minimal systems.

Abstract

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. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the $\omega_{\mathcal{F}}$-limit and the $\omega$-limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.