Indeterminacy loci of iterate maps in moduli space

TL;DR

This paper characterizes the points in the compactified moduli space of rational maps where the iteration map becomes indeterminate, shedding light on the structure of these indeterminacy loci in complex dynamics.

Contribution

It provides a complete characterization of the indeterminacy loci of the iterate maps in the moduli space of rational maps, extending understanding of their geometric and dynamical properties.

Findings

Identifies the elements in the compactification where iteration maps are indeterminate.

Provides criteria for indeterminacy in the moduli space.

Enhances understanding of the structure of iteration maps in complex dynamics.

Abstract

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