F-sensitivity and (\(F_1, F_2\))-sensitivity between dynamical systems and their induced hyperspace dynamical systems
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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.
Share and Cite
ISRP Style
Risong Li, Yu Zhao, Hongqing Wang, Ru Jiang, Haihua Liang, F-sensitivity and (\(F_1, F_2\))-sensitivity between dynamical systems and their induced hyperspace dynamical systems, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 1640--1651
AMA Style
Li Risong, Zhao Yu, Wang Hongqing, Jiang Ru, Liang Haihua, F-sensitivity and (\(F_1, F_2\))-sensitivity between dynamical systems and their induced hyperspace dynamical systems. J. Nonlinear Sci. Appl. (2017); 10(4):1640--1651
Chicago/Turabian Style
Li, Risong, Zhao, Yu, Wang, Hongqing, Jiang, Ru, Liang, Haihua. "F-sensitivity and (\(F_1, F_2\))-sensitivity between dynamical systems and their induced hyperspace dynamical systems." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 1640--1651
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
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