报告题目：On the dynamics of $H_{\lambda}(z)=z^{2}+\lambda \bar{z}$

　　报 告 人：张高飞

　　报告人所在单位：南京大学

　　报告日期：2020-08-26 星期三

　　报告时间：10:00-11:00

　　报告地点：腾讯会议ID: 331 711 300, 密码: 202020

　　报告摘要： It was found in 1990 s that the iteration of $H_{\lambda}$ for some specific $\lambda$ may produce very rich dynamics. For instance, Alexander and his collaborators showed that for $\lambda$ near $-(1+i 1.0287137)$, $H_{\lambda}$ has three attractors whose basins are intermingled in a very complicated fashion. Here we are going to study the dynamics of $H_{\lambda}$ by borrowing the ideas in complex dynamics. As an analogy to quadratic polynomials, we introduce the concept of filled-in Julia set for the family $H_{\lambda}$ and then define the connected locus by

　　\[M=\left\{\lambda \in \mathbb{C} \mid K_{\lambda} \text { is connected }\right\}.\]

　　As in complex dymamics, a fundamental question is to characterize the topology of $M .$ In this work we give a complete answer to this question. Let $C_{\lambda}=\{z \ | \ |z|=\frac{|\lambda|}{2}\}$ be the critical set of $H_{\lambda}$ and $T_{\lambda}=H_{\lambda}\left(C_{\lambda}\right)$ be the set ofsingular values which is a curved triangle. We will show that $M$ is the closure of a Jordan domain bounded by an algebraic curve consisting of all those $\lambda$ such that $K_{\lambda}$ is inscribed in $T_{\lambda}$