Finite Intersection Property and Dynamical Compactness

Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800–6827, 2016 ). In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynami...

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Published in:Journal of dynamics and differential equations Vol. 30; no. 3; pp. 1221 - 1245
Main Authors: Huang, Wen, Khilko, Danylo, Kolyada, Sergiĭ, Peris, Alfred, Zhang, Guohua
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2018
Springer Nature B.V
Subjects:
Applications of Mathematics
Chaos theory
Dynamical systems
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
Sensitivity
Dynamical topology
Dynamical compactness
Linear dynamics
Transitive sensitivity
Secondary 54H20
Multi-sensitivity
47A16
Sensitive compactness
Transitive compactness
Primary 37B05
Topological weak mixing
Hypercyclic operator
ISSN:1040-7294, 1572-9222
Online Access:Get full text
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Summary:Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800–6827, 2016 ). 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 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.
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ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-017-9600-8