Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2205–2222 | Cite as

Invariance Pressure Dimensions for Control Systems

  • Xingfu Zhong
  • Yu HuangEmail author


We introduce two kinds of invariance pressure dimension, which not only give dimension characteristics of invariance pressure and inner invariance pressure introduced by Colonius et al. (J Dyn Differ Equ,, 2018) for discrete-time control systems, but also generalize the corresponding concepts proposed by Huang and Zhong (Syst Control Lett 112:36–41, 2018). Then we derive some basic properties for them and investigate the relations among these invariance pressures defined by controls and by invariant covers. As an application, we characterize the complexity of a control system with zero invariance entropy.


Control system Invariance pressure Dimension characteristic Zero invariance entropy 



The authors are grateful to the referees for their valuable comments which have led to improvement of the paper. A part of this work was done when the first author visited Western University. The author would like to thank Prof. X. Zou and his institute for their hospitality.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsGuangdong University of Foreign StudiesGuangzhouPeople’s Republic of China
  2. 2.School of MathematicsSun Yat-Sen UniversityGuangzhouPeople’s Republic of China

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