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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1221–1245 | Cite as

Finite Intersection Property and Dynamical Compactness

  • Wen Huang
  • Danylo Khilko
  • Sergiĭ Kolyada
  • Alfred Peris
  • Guohua Zhang
Article
  • 132 Downloads

Abstract

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 \(\omega _{{\mathcal {F}}}\)-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.

Keywords

Dynamical topology Dynamical compactness Transitive compactness Sensitive compactness Topological weak mixing Multi-sensitivity Transitive sensitivity Linear dynamics Hypercyclic operator 

Mathematics Subject Classification

Primary 37B05 Secondary 54H20 47A16 

Notes

Acknowledgements

Wen Huang and Sergiĭ Kolyada acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai. Sergiĭ Kolyada also acknowledges the hospitality of the Max-Planck-Institute für Mathematik (MPIM) in Bonn, the Departament de Matemàtica Aplicada of the Universitat Politècnica de València, the partial support of Project MTM2013-47093-P, and the Department of Mathematics of the Chinese University of Hong Kong. We thank the referees for careful reading and constructive comments that have resulted in substantial improvements to this paper. Wen Huang was supported by NNSF of China (11225105, 11431012); Alfred Peris was supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P, and by GVA, Project PROMETEOII/2013/013; and Guohua Zhang was supported by NNSF of China (11671094).

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Wen Huang
    • 1
    • 2
  • Danylo Khilko
    • 3
  • Sergiĭ Kolyada
    • 4
  • Alfred Peris
    • 5
  • Guohua Zhang
    • 6
  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of MathematicsSichuan UniversityChengduChina
  3. 3.Département de Mathématiques et ApplicationsÉcole Normale SupŕieureParisFrance
  4. 4.Institute of Mathematics, NASUKyivUkraine
  5. 5.Institut Universitari de Matemàtica Pura i AplicadaUniversitat Politècnica de ValènciaValènciaSpain
  6. 6.School of Mathematical Sciences and LMNSFudan University and Shanghai Center for Mathematical SciencesShanghaiChina

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