From rubber bands to rational maps: A research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example

On Dynamics of Cubic Siegel Polynomials

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.Comment: 58 pages. 20 PostScript figure

Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps