Iterations of Meromorphic Functions involving Sine

Abstract

In this article, the dynamics of a one-parameter family of functions $f_{\hat{I}}»(z)=\frac{\sin z}{z^2+\hat{I}»},$ $\hat{I}»>0$, are studied. It shows the existence of parameters $0<\hat{I}{»}_1<\hat{I}{»}_2$ such that bifurcations occur at $\hat{I}{»}_1$ and $\hat{I}{»}_2$ for $f_{\hat{I}}»$. It is proved that the Fatou set $\mathcal{F}(f_{\hat{I}}»)$ is the union of basins of attraction in the complex plane for $\hat{I}»\in (\hat{I}{»}_1,\hat{I}{»}_2)\cup (\hat{I}{»}_2,\infty )$. Further, every Fatou component of $f_{\hat{I}}»$ is simply connected for $\hat{I}»\ge \hat{I}{»}_1$. The boundary of the Fatou set $\mathcal{F}(f_{\hat{I}}»)$ is the Julia set $\mathcal{J}(f_{\hat{I}}»)$ in the extended complex plane for $\hat{I}»>1$. Interestingly, it is found that $f_{\hat{I}}»$ has only one completely invariant Fatou component, say $U_{\hat{I}}»$ such that $\mathcal{F}(f_{\hat{I}}»)=U_{\hat{I}}»$ for $\hat{I}»>\hat{I}{»}_2$. Moreover, the characterization of the Julia set of $f_{\hat{I}}»$ is seen for $\hat{I}»\in (\hat{I}{»}_1,\infty )\setminus \{\hat{I}{»}_2\}$.