F-sensitivity and (\(F_1, F_2\))-sensitivity between dynamical systems and their induced hyperspace dynamical systems

Volume 10, Issue 4, pp 1640--1651 http://dx.doi.org/10.22436/jnsa.010.04.28

Publication Date: April 20, 2017 Submission Date: November 06, 2016

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Authors

Risong Li - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Yu Zhao - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Hongqing Wang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Ru Jiang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China. Haihua Liang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, P. R. China.

Abstract

The notions of \(F\)-sensitivity and (\(F_1, F_2\))-sensitivity were introduced and studied by Wang et al. via Furstenberg families in [H.-Y. Wang, J.-C. Xiong, F. Tan, Discrete Dyn. Nat. Soc., 2010 (2010), 12 pages]. In this paper, the concepts of \(F\)-collective sensitivity (resp. (\(F_1, F_2\))-collective sensitivity) and compact-type \(F\)-collective sensitivity (resp. compact-type (\(F_1, F_2\))-collective sensitivity) are introduced as stronger forms of the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively, where \(F,F_1\) and \(F_2\) are Furstenberg families. It is proved that \(F\)-sensitivity (resp. (\(F_1, F_2\))-sensitivity) of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the \(F\)-collective sensitivity (resp. (\(F_1, F_2\))-collective sensitivity) of the original system; F-sensitivity (resp. (\(F_1, F_2\))-sensitivity) of the induced hyperspace system defined on the space of all nonempty closed subsets (hit-or-miss topology) is equivalent to the compact-type \(F\)-collective sensitivity (resp. (\(F_1, F_2\))-collective sensitivity) of the original HLCSC system. Moreover, it is shown that for a given dynamical system (E, d, f) and a given Furstenberg family F, if (E, d, f) is F-mixing, then it is \(F\)-collectively sensitive. Additionally, we prove that for a given dynamical system (E, d, f) and a given Furstenberg family \(F, (E, d, f)\) is \(F\)-mixing if and only if \(\underbrace{f\times f\times...\times f}_n\) is \(F\)-mixing for every \(n\geq 2\). Our results extend and improve some existing results.

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Keywords

• Furstenberg families
• \(F\)-collective sensitivity
• compact-type \(F\)-collective sensitivity
• hyperspace dynamical systems
• compact-type (\(F_1، F_2\))-collective sensitivity
• (\(F_1،F_2\))-collective sensitivity
• hit-or-miss topology.

MSC

•  54H20
•  54B20
•  37D45

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