Complex Analysis, Dynamics and Geometry Seminar

In one-dimensional complex dynamics, we study the iterates of a rational function on the Riemann sphere. What happens when we replace the complex numbers with another field? This is peculiar when the norm/metric on the new field is non-Archimedean, one where disks can never partially overlap.

Remarkably, it turns out that we can trade the rational map for a piecewise-linear map of a real tree. Studying the dynamics on this tree has led to number theoretic results with $K = \mathbb Q_p$, and also theorems in one-dimensional complex dynamics when $K = \overline{\mathbb C(t)}$. In the latter case, one can consider the map as a two-dimensional map.

I will describe a new type of map, a skew product, which drops this restriction but maintains linearity on the tree. In this new setting, we recover the classification of Fatou components following J. Rivera-Letelier. Finally, I will give an application to the algebraic stability of complex rational skew products. Speaker(s): Richard Birkett (Notre Dame)