If F is Continuous on 2 2

Abstract

Let D be a dendrite and f be a continuous map on D. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points, the set of non-wandering points and the set of ω-limit points of x under f, respectively. Write \({\Omega _{k + 1}}\left( f \right) = \Omega \left( {f\left| {_{{\Omega _k}\left( f \right)}} \right.} \right)\) and \({\omega ^{k + 1}}\left( f \right) = \bigcup\nolimits_{x \in {\omega ^k}\left( f \right)} {\omega \left( {x,f} \right)} \) for any positive integer k, where Ω₁(f) = Ω(f) and \(\omega \left( f \right) = \bigcup\nolimits_{x \in D} {\omega \left( {x,f} \right)} \). ω ^m (f) is called the attracting centre of f if ω ^m+1(f)= ω ^m (f). In this paper, we show that if the rank of D is n − 1, then we have the following results: (1) ω ⁿ⁺²(f) = ω ⁿ⁺¹(f) and the attracting centre of f is ω ⁿ⁺¹(f); (2) \({\Omega _{n + 2}}\left( f \right) = \overline {R\left( f \right)} \) and the depth of f is at most n + 2. Further, if the set of (n − 1)-order accumulation points of Br(D) (the set of branch points of D) is a singleton, then \({\Omega _{n + 1}}\left( f \right) = \overline {R\left( f \right)} \) and the depth of f is at most n + 1. Besides, we show that there exist a dendrite D ₁ whose rank is n − 1 and the set of (n − 1)-order accumulation points of Br(D ₁) is a singleton, and a continuous map g on D ₁ such that ω ⁿ⁺¹(g) ≠ ω ⁿ (g) and \({\Omega _n}\left( f \right) \ne \overline {R\left( f \right)} \).

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Acknowledgements

We thank the referees for their careful reading of the manuscript and constructive comments and suggestions.

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Authors and Affiliations

1. Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Guangxi University of Finance and Economics, Nanning, 530003, P. R. China

   Guang Wang Su, Cai Hong Han, Tai Xiang Sun & Lue Li

2. College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, P. R. China

   Guang Wang Su, Cai Hong Han, Tai Xiang Sun & Lue Li

Corresponding author

Correspondence to Cai Hong Han.

Additional information

Supported NSF of Guangxi (Grant Nos. 2022GXNSFAA035552, 2020GXNSFAA297010) and PYMRBAP for Guangxi CU (Grant No. 2021KY0651)

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Su, G.W., Han, C.H., Sun, T.X. et al. The Depths and the Attracting Centres for Continuous Maps on a Dendrite Whose Rank is Finite. Acta. Math. Sin.-English Ser. 38, 1643–1652 (2022). https://doi.org/10.1007/s10114-022-0564-1

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• Received: 22 October 2020

• Revised: 26 October 2021

• Accepted: 13 December 2021

• Published: 15 September 2022

• Issue Date: September 2022

• DOI : https://doi.org/10.1007/s10114-022-0564-1

Keywords

• Dendrite
• attracting centre
• depth

MR(2010) Subject Classification

• 37E25
• 37B40
• 54H20

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