A study of a meromorphic perturbation of the sine family

Abstract

We study the dynamics of a meromorphic perturbation of the family λsinz\lambda \sin z by adding a pole at zero and a parameter μ\mu , that is, fλ,μ(z)=λsinz+μ/z{f}_{\lambda ,\mu }\left(z)=\lambda \sin z+\mu \hspace{-0.08em}\text{/}\hspace{-0.08em}z, where λ,μ∈C⧹{0}\lambda ,\mu \in {\mathbb{C}}\hspace{-0.16em}\setminus \hspace{-0.16em}\left\{0\right\}. We study some geometrical properties of fλ,μ{f}_{\lambda ,\mu } and prove that the imaginary axis is invariant under fn{f}^{n} and belongs to the Julia set when ∣λ∣≥1| \lambda | \ge 1. We give a set of parameters (λ,μ)\left(\lambda ,\mu ), such that the Fatou set of fλ,μ{f}_{\lambda ,\mu } has two super-attracting domains. If λ=1\lambda =1 and μ∈(0,2)\mu \in \left(0,2), the Fatou set of f1,μ{f}_{1,\mu } has two attracting domains. Also, we give parameters λ,μ\lambda ,\mu such that ±π/2\pm \pi \hspace{-0.08em}\text{/}\hspace{-0.08em}2 are fixed points of fλ,μ{f}_{\lambda ,\mu } and the Fatou set of fλ,μ{f}_{\lambda ,\mu } contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of fλ,μ{f}_{\lambda ,\mu }, where the Fatou set contains two types of domains, for λ,μ\lambda ,\mu given.

Keywords

• iteration
• fixed points
• fatou set
• julia set
• 37f10
• 30d05