Faculty Publications
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Item Relation between kneading matrices of a map and its iterates(Korean Mathematical Society kms@kms.or.kr, 2020) Gopalakrishna, C.; Murugan, V.It is known that the kneading matrix associated with a continuous piecewise monotone self-map of an interval contains crucial combinatorial information of the map and all its iterates, however for every iterate of such a map we can associate its kneading matrix. In this paper, we describe the relation between kneading matrices of maps and their iterates for a family of chaotic maps. We also give a new definition for the kneading matrix and describe the relationship between the corresponding determinant and the usual kneading determinant of such maps. ©2020 Korean Mathematical Society.Item 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.Item Invariance of kneading matrix under conjugacy(Korean Mathematical Society, 2021) Gopalakrishna, C.; Veerapazham, M.In the kneading theory developed by Milnor and Thurston, it is proved that the kneading matrix and the kneading determinant associated with a continuous piecewise monotone map are invariant under orientation-preserving conjugacy. This paper considers the problem for orientation-reversing conjugacy and proves that the former is not an invariant while the latter is. It also presents applications of the result towards the computational complexity of kneading matrices and the clas-sification of maps up to topological conjugacy. © 2021 Korean Mathematial Soiety.Item Existence, construction and extension of continuous solutions of an iterative equation with multiplication(Science Press (China), 2023) Gopalakrishna, C.; Veerapazham, M.; Wang, S.; Zhang, W.The iterative equation is an equality with an unknown function and its iterates, most of which found from references are a linear combination of those iterates. In this paper, we work on an iterative equation with multiplication of iterates of the unknown function. First, we use an exponential conjugation to reduce the equation on ℝ+ to the form of the linear combination on ℝ, but those known results on the linear combination were obtained on a compact interval or a neighborhood near a fixed point. We use the Banach contraction principle to give the existence, uniqueness and continuous dependence of continuous solutions on ℝ+ that are Lipschitzian on their ranges, and construct its continuous solutions on ℝ+ sewing piece by piece. We technically extend our results on ℝ+ to ℝ− and show that none of the pairs of solutions obtained on ℝ+ and ℝ− can be combined at the origin to get a continuous solution of the equation on the whole ℝ, but can extend those given on ℝ+ to obtain continuous solutions on the whole ℝ. © 2023, Science China Press.Item 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.