Schedule for: 25w5345 - Holomorphic Dynamics

A Kleinian group is a discrete subgroup of Moebius transformations in $PSL(2,C)$. These naturally act as isometries of hyperbolic three-space. But they also act naturally on the Riemann sphere. There are many parallels between the dynamics of this action and the dynamics of iteration of rational functions, which gives rise to Sullivan's dictionary relating them. In the first lecture I will talk about the action of Kleinian groups on the Riemann sphere, which can be divided into two sets: an open set where the group acts properly discontinuously and a closed set where orbits are dense. These sets are the analogue of the Fatou and Julia sets for iterated rational functions. We will discuss the possible types of set that can arise in this way. In the second lecture, I will discuss two parametrised families of Kleinian groups and we will see how the dynamics change as the parameters vary. This gives rise to sets that are analogous to the Mandelbrot set.

A Kleinian group is a discrete subgroup of Moebius transformations in $PSL(2,C)$. These naturally act as isometries of hyperbolic three-space. But they also act naturally on the Riemann sphere. There are many parallels between the dynamics of this action and the dynamics of iteration of rational functions, which gives rise to Sullivan's dictionary relating them. In the first lecture I will talk about the action of Kleinian groups on the Riemann sphere, which can be divided into two sets: an open set where the group acts properly discontinuously and a closed set where orbits are dense. These sets are the analogue of the Fatou and Julia sets for iterated rational functions. We will discuss the possible types of set that can arise in this way. In the second lecture, I will discuss two parametrised families of Kleinian groups and we will see how the dynamics change as the parameters vary. This gives rise to sets that are analogous to the Mandelbrot set.