Indeterminacy Loci of Iterate Maps in Moduli Space

Date
2019/12/13 Fri 14:30 - 17:00
Room
6号館609号室
Speaker
Jan Kiwi
Affiliation
Pontificia Universidad Católica de Chile
Abstract

We study the action of the iteration maps on moduli spaces of complex rational maps. The tools employed emerge from considering dynamical systems acting on the Berkovich projective line over an appropriate non-Archimedean field.

The moduli space $\operatorname{rat}_d$ of rational maps in one complex variable of degree $d \ge 2$ has a natural compactification by a projective variety $\overline{\operatorname{rat}}_d$ provided by geometric invariant theory. Given $n \ge 2$, the iteration map $\Phi_n : \operatorname{rat}_d \to \operatorname{rat}_{d^n}$, defined by $\Phi_n:
[f] \mapsto [f^n]$, extends to a rational map $\Phi_n: \overline{\operatorname{rat}}_d \dashrightarrow\overline{\operatorname{rat}}_{d^n}$. We characterize the elements of $\overline{\operatorname{rat}}_d$ which lie in the indeterminacy locus of $\Phi_n$.
This is a joint work with Hongming Nie (Hebrew University of Jerusalem).