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).