Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 6, pp 1605–1618 | Cite as

Chain Mixing, Shadowing Properties and Multi-transitivity

  • Huoyun Wang
  • Heman FuEmail author
  • Sulan Diao
  • Peng Zeng
Original Paper


We prove that a map f is chain mixing if and only if \(f^r\times f^s\) is chain transitive for some positive integers rs. We prove that a map which has the average shadowing property with dense \(\underline{0}\)-recurrent points is transitive, and by this result we point out that a map is multi-transitive if it has the average shadowing property and an invariant Borel probability measure with full support. Moreover, we show that \(\Delta \)-mixing, the completely uniform positive entropy and the average shadowing property are equivalent mutually for a surjective map which has the shadowing property.


Shadowing property Average shadowing Multi-transitivity Completely uniform positive entropy Chain mixing 

Mathematics Subject Classification

37B05 34D05 



This research was supported by National Nature Science Funds of China (Grant Nos. 11471125, 11771149). The authors would like to thank the referees for their helpful suggestions and remarks.


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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsGuangzhou UniversityGuangzhouPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsZhaoqing UniversityZhaoqingPeople’s Republic of China

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