# A quantitative approach to syndetic transitivity and topological ergodicity

Volume 10, Issue 9, pp 4680--4686
Publication Date: September 08, 2017 Submission Date: October 30, 2016
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### Authors

Yu Zhao - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Risong Li - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Tianxiu Lu - Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People's Republic of China. - Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, People’s Republic of China. - Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province, Zigong, Sichuan, 643000, People’s Republic of China. Ru Jiang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Hongqing Wang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China. Haihua Liang - School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, 524025, People's Republic of China.

### Abstract

In this paper, we give new quantitative characteristics of degrees of syndetical transitivity and topological ergodicity for a given discrete dynamical system, which are nonnegative real numbers and are not more than $1$. For selfmaps of many compact metric spaces it is proved that a given selfmap is syndetically transitive if and only if its degree of syndetical transitivity is $1$, and that it is topologically ergodic if and only if its degree of topological ergodicity is one. Moreover, there exists a selfmap of $[0, 1]$ having all degrees positive.

### Share and Cite

##### ISRP Style

Yu Zhao, Risong Li, Tianxiu Lu, Ru Jiang, Hongqing Wang, Haihua Liang, A quantitative approach to syndetic transitivity and topological ergodicity, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4680--4686

##### AMA Style

Zhao Yu, Li Risong, Lu Tianxiu, Jiang Ru, Wang Hongqing, Liang Haihua, A quantitative approach to syndetic transitivity and topological ergodicity. J. Nonlinear Sci. Appl. (2017); 10(9):4680--4686

##### Chicago/Turabian Style

Zhao, Yu, Li, Risong, Lu, Tianxiu, Jiang, Ru, Wang, Hongqing, Liang, Haihua. "A quantitative approach to syndetic transitivity and topological ergodicity." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4680--4686

### Keywords

• Sensitivity
• syndetically sensitive
• ergodically sensitive
• multi-sensitive
• cofinitely sensitive
• Furstenberg families.

•  37B10
•  37C20
•  37C50

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