标题： 亚纯函数迭代中涉及正弦的研究 显示英文标题

标题： Iterations of Meromorphic Functions involving Sine

摘要： 本文研究了一参数函数族$f_{\lambda}(z) = \frac{\sin{z}}{z^2 + \lambda},$ $\lambda>0$ 的动力学性质。它展示了存在参数$0< \lambda_{1}< \lambda_{2}$使得在$\lambda_1$和$\lambda_2$处对于$f_{\lambda}$发生了分岔。 已经证明，Fatou集$\mathcal{F}(f_{\lambda})$是复平面上吸引域的并集，对于$\lambda \in (\lambda_1, \lambda_2) \cup (\lambda_2, \infty)$来说成立。 此外，对于$\lambda \geq \lambda_1$，$f_{\lambda}$的每一个Fatou分支都是单连通的。 Fatou集$\mathcal{F}(f_{\lambda})$的边界是扩展复平面上的Julia集$\mathcal{J}(f_{\lambda})$，对于$\lambda> 1$来说成立。 有趣的是，发现$f_{\lambda}$仅有一个完全不变的Fatou分支，记作$U_\lambda$满足$\mathcal{F}(f_{\lambda}) = U_{\lambda}$对于$\lambda >\lambda_2$。此外，$f_{\lambda}$的Julia集的特征在$\lambda \in (\lambda_1, \infty)\setminus \{\lambda_2\}$下被观察到。 显示英文摘要

摘要： 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 $\lambda \in (\lambda_1, \infty)\setminus \{\lambda_2\}$.