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Finite intersection property and dynamical compactness

Wen Huang, Danylo Khilko, Sergiy Kolyada, Alfred Peris, Guohua Zhang

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 $ω_{\mathcal{F}}$-limit and the $ω$-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.

Finite intersection property and dynamical compactness

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 -limit and the -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.

Paper Structure

This paper contains 11 sections, 27 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concepts in topological dynamics
  4. Basic concepts of Furstenberg families
  5. The concept of an almost one-to-one map
  6. Symbolic dynamics
  7. Dynamical compactness with respect to an arbitrary family
  8. Transitive sensitivity and sensitive compactness
  9. Weakly disjointness and weakly mixing
  10. Transitive compact (non weakly mixing) systems
  11. Weak transitive compactness and sensitivity for linear operators

Key Result

Proposition 2.1

Let $x\in X$. Then $T \omega_\mathcal{F} (x)\subset \omega_\mathcal{F} (T x)$. Additionally, if $\mathcal{F}$ is $-$ ($+$, translation, respectively) invariant then $\omega_\mathcal{F} (T x)\subset$ ($\supset$, $=$, respectively) $\omega_\mathcal{F} (x)$.

Theorems & Definitions (54)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 44 more