A quantitative approach to syndetic transitivity and topological ergodicity


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.


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