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Author: search.filters.author.Gopalakrishna, C.Subject: search.filters.subject.Iteration operator

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    Dynamics of the iteration operator on the space of continuous self-maps
    (American Mathematical Society, 2021) Murugan, M.; Gopalakrishna, C.; Zhang, W.
    The semi-dynamical system of a continuous self-map is generated by iteration of the map, however, the iteration itself, being an operator on the space of continuous self-maps, may generate interesting dynamical behaviors. In this paper we prove that the iteration operator is continuous on the space of all continuous self-maps of a compact metric space and therefore induces a semi-dynamical system on the space. Furthermore, we characterize its fixed points and periodic points in the case that the compact metric space is a compact interval by discussing the Babbage equation. We prove that all orbits of the iteration operator are bounded but most fixed points are not stable. On the other hand, we prove that the iteration operator is not chaotic. © 2020 American Mathematical Society.
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    Dynamics of iteration operators on self-maps of locally compact Hausdorff spaces
    (Cambridge University Press, 2024) Gopalakrishna, C.; Veerapazham, M.; Zhang, W.
    In this paper, we prove the continuity of iteration operators on the space of all continuous self-maps of a locally compact Hausdorff space X and generally discuss dynamical behaviors of them. We characterize their fixed points and periodic points for and the unit circle. Then we indicate that all orbits of are bounded; however, we prove that for and, every fixed point of which is non-constant and equals the identity on its range is not Lyapunov stable. The boundedness and the instability exhibit the complexity of the system, but we show that the complicated behavior is not Devaney chaotic. We give a sufficient condition to classify the systems generated by iteration operators up to topological conjugacy. © The Author(s), 2023. Published by Cambridge University Press.