Sensitivity for topologically double ergodic dynamical systems

Risong Li; Tianxiu Lu; Xiaofang Yang; Yongxi Jiang; Risong Li; Tianxiu Lu; Xiaofang Yang; Yongxi Jiang

doi:10.3934/math.2021609

AIMS Mathematics

2021, Volume 6, Issue 10: 10495-10505. doi: 10.3934/math.2021609

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Research article

Sensitivity for topologically double ergodic dynamical systems

1.
School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China
2.
College of Mathematics and Statistics, Sichuan University of Science and Engineering; The Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, Sichuan 643000, China
3.
Artificial Intelligence Key Laboratory of Sichuan Province; Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province, Zigong, Sichuan 643000, China

Received: 23 February 2021 Accepted: 15 July 2021 Published: 20 July 2021
MSC : 54H20, 37B45, 37C20, 37C50

As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.
- sensitivity,
- multi-sensitivity,
- syndetic sensitivity,
- ergodic (resp. strongly ergodic) multi-sensitivity
Citation: Risong Li, Tianxiu Lu, Xiaofang Yang, Yongxi Jiang. Sensitivity for topologically double ergodic dynamical systems[J]. AIMS Mathematics, 2021, 6(10): 10495-10505. doi: 10.3934/math.2021609

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Abstract

As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.

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