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On Limit Sets of Monotone Maps on Regular Curves

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Abstract

We investigate the structure of \(\omega \)-limit (resp. \(\alpha \)-limit) sets for a monotone map f on a regular curve X. We show that for any \(x\in X\) (resp. for any negative orbit \((x_{n})_{n\ge 0}\) of x), the \(\omega \)-limit set \(\omega _{f}(x)\) (resp. \(\alpha \)-limit set \(\alpha _{f}((x_{n})_{n\ge 0})\)) is a minimal set. This also holds for \(\alpha \)-limit set \(\alpha _{f}(x)\) whenever x is not a periodic point. These results extend those of Naghmouchi [24] established whenever f is a homeomorphism on a regular curve and those of Abdelli [1], whenever f is a monotone map on a local dendrite. Further results related to the basin of attraction of an infinite minimal set are also obtained.

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Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions. This work was supported by the research unit: “Dynamical systems and their applications”, (UR17ES21), Ministry of Higher Education and Scientific Research, Faculty of Science of Bizerte, Bizerte, Tunisia.

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  1. Faculty of Science of Bizerte, (UR17ES21), “Dynamical systems and their applications”, University of Carthage, Jarzouna, Bizerte, 7021, Tunisia

    Aymen Daghar & Habib Marzougui

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  1. Aymen Daghar
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  2. Habib Marzougui
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Correspondence to Habib Marzougui.

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Daghar, A., Marzougui, H. On Limit Sets of Monotone Maps on Regular Curves. Qual. Theory Dyn. Syst. 20, 89 (2021). https://doi.org/10.1007/s12346-021-00523-4

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