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Homeomorphism of Hereditarily Locally Connected Continua

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Abstract

Let \(f:X\rightarrow X\) be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) and \(\Omega (f)\), the sets of periodic points, almost periodic points and the non-wandering points of f. We show that any \(\omega \)-limit set (resp. \(\alpha \)-limit set) is minimal. Moreover, we show that \(\Omega (f)=AP(f)\). We also prove that if \(P(f)=\emptyset \), then there exists a unique minimal set. On the other hand, if \(P(f)\ne \emptyset \) then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum.

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Acknowledgements

The author expresses many thanks to the referee for useful suggestions. The author is thankful to Mr.Issam Naghmouchi for his helpful remarks and discussion during the preparation of the paper.

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  1. Faculty of Sciences of Bizerte, (UR17ES21), “Dynamical Systems and their Applications”, University of Carthage, 7021, Jarzouna, Tunisia

    Aymen Daghar

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Daghar, A. Homeomorphism of Hereditarily Locally Connected Continua. J Dyn Diff Equat 35, 3269–3293 (2023). https://doi.org/10.1007/s10884-021-10064-8

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  • DOI: https://doi.org/10.1007/s10884-021-10064-8

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