## 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.## References

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## Bibliographic Information

**Laura DeMarco**- Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
- MR Author ID: 677013
- Email: demarco@math.uchicago.edu
- Received by editor(s): February 28, 2005
- Published electronically: February 16, 2006
- Additional Notes: Research was partially supported by an NSF Postdoctoral Fellowship
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**20**(2007), 321-355 - MSC (2000): Primary 37F45; Secondary 14L24, 57M50
- DOI: https://doi.org/10.1090/S0894-0347-06-00527-3
- MathSciNet review: 2276773