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The moduli space of quadratic rational maps


Author: Laura DeMarco
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
Published electronically: February 16, 2006
MathSciNet review: 2276773
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Abstract: Let $ M_2$ be the space of quadratic rational maps $ f:{\bf P}^1\to {\bf 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{\bf 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 $ {\bf 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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Additional Information

Laura DeMarco
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: demarco@math.uchicago.edu

DOI: https://doi.org/10.1090/S0894-0347-06-00527-3
Received by editor(s): February 28, 2005
Published electronically: February 16, 2006
Additional Notes: Research was partially supported by an NSF Postdoctoral Fellowship
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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