## The moduli space of quadratic rational maps

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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

- Laura DeMarco,
*Iteration at the boundary of the space of rational maps*, Duke Math. J.**130**(2005), no. 1, 169–197. MR**2176550**, DOI 10.1215/S0012-7094-05-13015-0 - Adrien Douady and Clifford J. Earle,
*Conformally natural extension of homeomorphisms of the circle*, Acta Math.**157**(1986), no. 1-2, 23–48. MR**857678**, DOI 10.1007/BF02392590 - Igor Dolgachev,
*Lectures on invariant theory*, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR**2004511**, DOI 10.1017/CBO9780511615436 - Adam Lawrence Epstein,
*Bounded hyperbolic components of quadratic rational maps*, Ergodic Theory Dynam. Systems**20**(2000), no. 3, 727–748. MR**1764925**, DOI 10.1017/S0143385700000390 - Ricardo Mañé,
*On the uniqueness of the maximizing measure for rational maps*, Bol. Soc. Brasil. Mat.**14**(1983), no. 1, 27–43. MR**736567**, DOI 10.1007/BF02584743 - John Hubbard, Peter Papadopol, and Vladimir Veselov,
*A compactification of Hénon mappings in $\textbf {C}^2$ as dynamical systems*, Acta Math.**184**(2000), no. 2, 203–270. MR**1768111**, DOI 10.1007/BF02392629 - M. Ju. Ljubich,
*Entropy properties of rational endomorphisms of the Riemann sphere*, Ergodic Theory Dynam. Systems**3**(1983), no. 3, 351–385. MR**741393**, DOI 10.1017/S0143385700002030 - Ricardo Mañé,
*On the uniqueness of the maximizing measure for rational maps*, Bol. Soc. Brasil. Mat.**14**(1983), no. 1, 27–43. MR**736567**, DOI 10.1007/BF02584743 - Ricardo Mañé,
*The Hausdorff dimension of invariant probabilities of rational maps*, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86–117. MR**961095**, DOI 10.1007/BFb0083068 - John Milnor,
*Geometry and dynamics of quadratic rational maps*, Experiment. Math.**2**(1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR**1246482**, DOI 10.1080/10586458.1993.10504267 - D. Mumford, J. Fogarty, and F. Kirwan,
*Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR**1304906**, DOI 10.1007/978-3-642-57916-5 - Joseph H. Silverman,
*The space of rational maps on $\mathbf P^1$*, Duke Math. J.**94**(1998), no. 1, 41–77. MR**1635900**, DOI 10.1215/S0012-7094-98-09404-2

## 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