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Dynamics of a family of meromorphic functions with two essential singularities which are not omitted values. (English) Zbl 1459.37033

Summary: 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.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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