Iterations of Meromorphic Functions involving Sine

TL;DR

The paper analyzes the iteration dynamics of the meromorphic family $f_{\lambda}(z)=\frac{\sin z}{z^{2}+\lambda}$, $\lambda>0$. It identifies two bifurcation parameters $0<\lambda_{1}<\lambda_{2}$ and characterizes the Fatou and Julia sets across parameter ranges, showing that for $\lambda\in(\lambda_{1},\lambda_{2})\cup(\lambda_{2},\infty)$ the Fatou set is the union of attracting basins, with all Fatou components simply connected for $\lambda\geq\lambda_{1}$ and a unique completely invariant Fatou component for $\lambda>\lambda_{2}$. It proves a connected Julia set for $\lambda\geq\lambda_{1}$ and provides a Julia-set characterization $\mathcal{J}(f_{\lambda})=\overline{I(f_{\lambda})}$ for $\lambda\in(\lambda_{1},\infty)\setminus\{\lambda_{2}\}$, while ruling out Baker/ wandering domains in these regimes. The work reveals rich, parameter-dependent dynamics, including real and imaginary-axis bifurcations, and sets the stage for deeper exploration of the $\lambda\to0$ limit.

Abstract

In this article, the dynamics of a one-parameter family of functions $f_λ(z) = \frac{\sin{z}}{z^2 + λ},$ $λ>0$, are studied. It shows the existence of parameters $0< λ_{1}< λ_{2}$ such that bifurcations occur at $λ_1$ and $λ_2$ for $f_λ$. It is proved that the Fatou set $\mathcal{F}(f_λ)$ is the union of basins of attraction in the complex plane for $λ\in (λ_1, λ_2) \cup (λ_2, \infty)$. Further, every Fatou component of $f_λ$ is simply connected for $λ\geq λ_1$. The boundary of the Fatou set $\mathcal{F}(f_λ)$ is the Julia set $\mathcal{J}(f_λ)$ in the extended complex plane for $λ> 1$. Interestingly, it is found that $f_λ$ has only one completely invariant Fatou component, say $U_λ$ such that $\mathcal{F}(f_λ) = U_λ$ for $λ>λ_2$. Moreover, the characterization of the Julia set of $f_λ$ is seen for $λ\in (λ_1, \infty)\setminus \{λ_2\}$.

Figures (5)

• Figure 1: The graphs of $g_{\lambda}(x)$ for (a) $\lambda^*< \lambda < 1, \text{(b)} \; \lambda= 1$ and $\text{(c)} \;\lambda > 1$.
• Figure 2: The graphs of $f^2_{\lambda}(x)$ for (a) $\lambda^*< \lambda < \hat{\lambda}, \text{(b)} \; \lambda= \lambda^*$ and $\text{(c)} \;\lambda < \lambda^*$.
• Figure 3: The graphs of $g_{\lambda}(y)=-h_{\lambda}(y)-y$ for (a) $\lambda= \lambda_1,$ (b) $\lambda_1< \lambda < \lambda_2,$ (c) $\lambda= \lambda_2$ and $\text{(d)} \;\lambda > \lambda_2$.
• Figure 4: Red colour region represents the Fatou set of $f_{\lambda}$ for $\lambda_{1}< \lambda< \lambda_{2}$. Here $-1.5 \pi <x< 1.5 \pi$ and $-2 \pi< y <0$.
• Figure 5: Red color region represents the Fatou set of $f_{\lambda}$ for $\lambda> \lambda_{2}$. Here $-1.5 \pi <x< 1.5 \pi$ and $-2 \pi< y <0$.

Theorems & Definitions (38)

• Proposition 1
• proof
• Lemma 2
• proof
• Proposition 3
• proof
• Corollary 4
• Proposition 5
• proof
• Proposition 6