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

• Domínguez, P. ^[1] ; Sienra, G. ^[2] ; Hernández, I. ^[1]
  1. [1] Benemérita Universidad Autónoma de Puebla

     Benemérita Universidad Autónoma de Puebla

     México

     
  2. [2] UNAM (México)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 1, 2021
• Idioma: inglés
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• Resumen

  • In this article we investigate the dynamics of the family Fλ,c,μ(z) = λe1/(z2+c) + μ, where λ, c ∈ C\{0} and μ ∈ C\{±i √c}, with two essential singularities which are not omitted values. Choosing a slice of the space of parameters, we prove that for certain parameters λ, c and μ, 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 (Fλ,c,μ, U, V ) is a polynomial-like mapping of degree two for certain values of the parameters λ, c, μ, and some domains U and V. Also, some examples of the Fatou and Julia sets for the polynomial-like mapping are given.