Iteration at the boundary of the space of rational maps
By Laura Demarco. 2004
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Let Ratd denote the space of holomorphic self-maps of P of degree d ≥ 2, and let μf be the measure of maximal entropy for f ∈ Ratd . The map of measures f → μf is known to be continuous on Ratd , and it is shown here to extend continuously to the boundary of Ratd in Ratd P H(P1 × P1,O(d, 1)) P2d+1, except along a locus I (d) of codimension d + 1. The set I (d) is also the indeterminacy locus of the iterate map f → f n for every n ≥ 2. The limiting measures are given explicitly, away from I (d). The degenerations of rational maps are also described in terms of metrics of nonnegative curvature on the Riemann sphere; the limits are polyhedral. 0. Introduction For each integer d ≥ 1, let Ratd denote the space of holomorphic maps f : P1 → P1 of degree d with the topology of uniform convergence. Fixing a coordinate system on the projective line, each such map can be expressed as a ratio of homogeneous polynomials f (z : w) = (P (z,w) : Q(z,w)), where P and Q have no common factors and are both of degree d. Parametrizing the space Ratd by the coefficients of P and Q, we have Ratd P2d+1\V (Res), where V (Res) is the hypersurface of polynomial pairs (P,Q) for which the resultant vanishes. In particular, Ratd is smooth and affine. In this article, we aim to describe the possible limiting behavior of a sequence of rational maps which diverges in Ratd for each d ≥ 2 in terms of the measures of maximal entropy and corresponding conformal metrics on the Riemann sphere. This is the first step in describing a dynamically natural compactification of this space or a boundary of the moduli space Ratd/PSL2 C, where the group of Mobius transformations acts by conjugation on Ratd . A compactification of the moduli space has been studied by Milnor [Mi] and Epstein [E] in degree 2 and Silverman [S] in all DUKE MATHEMATICAL JOURNAL Vol. 130, No. 1, c © 2005 Received 2 March 2004. Revision received 9 February 2005. 2000 Mathematics Subject Classification. Primary 37F45. Research partially supported by a National Science Foundation postdoctoral fellowship.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
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Citations in the corpus
- The moduli space of quadratic rational maps (2004, unknown): “ of M2 in M2 ≃ P2 by successively attaching 2-spheres at a countable collection of points in M2, as in Figure 1. Outline of the paper. The notation is fixed in Section 2, and a summary of results from [De] is provided. The geometric invariant theory compactification Md is defined in Section 3. In Secti”
- The Space of Morphisms on Projective Space (2009, methodology): “In [3], DeMarco uses the explicit description of the space M2 in order to study iterations of quadratic maps on P1, and one can expect similar results in higher dimension given a better understanding of the structure of Mnd .”