Iterations of Meromorphic Functions involving Sine

TL;DR

This paper explores the complex dynamics of a family of meromorphic functions involving sine, identifying bifurcations, Fatou and Julia sets, and invariant components as the parameter varies.

Contribution

It provides a detailed analysis of the bifurcation structure and Fatou-Julia set properties for the family of functions involving sine and a parameter.

Findings

Existence of bifurcation parameters $\lambda_1$ and $\lambda_2$.

Fatou set is union of basins of attraction for certain parameter ranges.

Unique invariant Fatou component for $\lambda > \lambda_2$.

Abstract

In this article, the dynamics of a one-parameter family of functions $f_{\lambda}(z) = \frac{\sin{z}}{z^2 + \lambda},$ $\lambda>0$, are studied. It shows the existence of parameters $0< \lambda_{1}< \lambda_{2}$ such that bifurcations occur at $\lambda_1$ and $\lambda_2$ for $f_{\lambda}$. It is proved that the Fatou set $\mathcal{F}(f_{\lambda})$ is the union of basins of attraction in the complex plane for $\lambda \in (\lambda_1, \lambda_2) \cup (\lambda_2, \infty)$. Further, every Fatou component of $f_{\lambda}$ is simply connected for $\lambda \geq \lambda_1$. The boundary of the Fatou set $\mathcal{F}(f_{\lambda})$ is the Julia set $\mathcal{J}(f_{\lambda})$ in the extended complex plane for $\lambda> 1$. Interestingly, it is found that $f_{\lambda}$ has only one completely invariant Fatou component, say $U_\lambda$ such that $\mathcal{F}(f_{\lambda}) = U_{\lambda}$ for $\lambda >\lambda_2$. Moreover, the characterization of the Julia set of $f_{\lambda}$ is seen for $\lambda \in (\lambda_1, \infty)\setminus \{\lambda_2\}$.