The moduli space of quadratic rational maps

DeMarco, Laura

doi:10.1090/S0894-0347-06-00527-3

The moduli space of quadratic rational maps
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by Laura DeMarco

J. Amer. Math. Soc. 20 (2007), 321-355

DOI: https://doi.org/10.1090/S0894-0347-06-00527-3

Published electronically: February 16, 2006

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Abstract:

Let $M_2$ be the space of quadratic rational maps $f:\textbf {P}^1\to \textbf {P}^1$, modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification $X$ of $M_2$ is defined, as a modification of Milnor’s $\overline {M}_2\simeq \textbf {CP}^2$, by choosing representatives of a conjugacy class $[f]\in M_2$ such that the measure of maximal entropy of $f$ has conformal barycenter at the origin in $\textbf {R}^3$ and taking the closure in the space of probability measures. It is shown that $X$ is the smallest compactification of $M_2$ such that all iterate maps $[f]\mapsto [f^n]\in M_{2^n}$ extend continuously to $X \to \overline {M}_{2^n}$, where $\overline {M}_d$ is the natural compactification of $M_d$ coming from geometric invariant theory.

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