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Dynamics of a Family of Meromorphic Functions with Two Essential Singularities Which Are Not Omitted Values

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Abstract

In this article we investigate the dynamics of the family \(F_{\lambda ,c,\mu }(z)= \lambda e^{1/(z^2+c)} + \mu \), where \(\lambda ,c\in {\mathbb C}{\setminus }\{0\}\) and \(\mu \in {\mathbb C}{\setminus }\{\pm i\sqrt{c}\}\), with two essential singularities which are not omitted values. Choosing a slice of the space of parameters, we prove that for certain parameters \(\lambda ,c\) and \(\mu \), the Fatou set contains a completely invariant and multiply connected attracting domain, a parabolic domain and a Siegel disc. Moreover, we prove that the triple \((F_{\lambda ,c,\mu }, U, V)\) is a polynomial-like mapping of degree two for certain values of the parameters \(\lambda \), c, \(\mu \), and some domains U and V. Also, some examples of the Fatou and Julia sets for the polynomial-like mapping are given.

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Acknowledgements

We would like to thank to the referee for the careful reading of the article and many helpful suggestions. The paper is dedicated to the memory of Iván Hernández, third author, who passed away while the paper was under revision. This research was supported by project PAPIIT IN106719.

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  1. Fac. Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, C.U., 72570, Puebla Pue, Mexico

    P. Domínguez & I. Hernández

  2. Facultad de Ciencias, UNAM, Av. Universidad 3000, C.U., 04510, Mexico, Mexico

    G. Sienra

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  1. P. Domínguez
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  2. G. Sienra
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  3. I. Hernández
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Correspondence to P. Domínguez.

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Domínguez, P., Sienra, G. & Hernández, I. Dynamics of a Family of Meromorphic Functions with Two Essential Singularities Which Are Not Omitted Values. Qual. Theory Dyn. Syst. 20, 1 (2021). https://doi.org/10.1007/s12346-020-00443-9

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  • DOI: https://doi.org/10.1007/s12346-020-00443-9

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