A Study of the Dynamics of the Family f λ ,μ = λsinz+μ/(z−kπ) where λ ,μ ∈ R\{0} and k ∈ Z\{0}
DOI:
https://doi.org/10.5890/DNC.2016.12.006Abstract
In this article we investigate the dynamics of the meromorphic family f λ ,μ (z) = λ sin z+ μ/( z−kπ) , λ ,μ ∈ R \ {0} and k ∈ Z \ {0}. We show that for some parameters λ ,μ the Stable set contains an attracting component which is multiply connected and completely invariant. We give a definition of a cut of the space of parameters, with μ and kπ fixed, and show examples of a cut and the Stable and Chaotic sets related to the cut, for some λ given.References
[1] Fatou, P. (1926), Sur L' itération des Fonctions Transcendentes Entierès, Acta Math., 47, 337-370.
[2] Baker, I.N. (1959), Fix Points and Iterates of Entire Functions, Math. Z., 71, 146-153.
[3] Baker, I. N. (1975), The Domains of Normality of an Entire Funtion, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 1, 277-283.
[4] Devaney, B. (1984), Julia Sets and Bifurcation Diagrams for ExponentialMaps, Bulletin of the AmericanMathematical Society, 11, 167-172.
[5] Bhattacharyya, P. (1969), Iteration of Analytic Functions. PhD Thesis, University of London.
[6] Domínguez, P. and Sienra, G. (2002), A Study of the Dynamics of the Family λsenz, International Journal of Bifurcation and Chaos, 12, 2869-2883.
[7] Baker, I. N., Kotus J. and Yinian Lü. (1991), Iterates ofMeromorphic Functions I, Ergodic Theory Dynamical Systems, 11, 241-248.
[8] Baker, I.N., Kotus J. and Yinian Lü. (1990), Iterates of Meromorphic Functions II: Examples of Wandering Domains, J. London. Math. Soc., 42, 267-278.
[9] Baker, I.N., Kotus J. and Yinian Lü. (1991), Iterates of Meromorphic Functions III: Preperiodic Domains, Ergodic Theory Dynamical Systems, 11 (2), 603-618.
[10] Baker, I.N., Kotus J. and Yinian Lü. (1992), Iterates ofMeromorphic Functions IV: Critically Finite Functions, Results in Mathematics, 22, 651-656.
[11] Domínguez, P. (1998), Dynamics of Transcendental Meromorphic Functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 23, 225-250.
[12] Herring, M.E. (1994), An Extension of the Julia-Fatou Theory of Iteration, PhD Thesis, University of London.
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