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MiS Preprint
74/2003

Equivariant Rational maps and Configurations: spherical equidistribution and SO(N,1) contraction

Sidney Frankel

Abstract

We build up a class of O(N,1)-intrinsic spherical rational maps, using only stereographic projections and affine centers of mass, and slightly extend it with antipodal maps. The geometric-analysis of their dynamics lends itself to applications to equidistribution of points on the sphere and to canonical global parametrizations of the rational maps of ${\rm \mathbb{C}P}^1$. We construct geometrically natural examples of rational maps of ${\rm \mathbb{C}P}^n$, and introduce a new approach, "suspension", to producing iterative algorithms for factoring polynomials, and to finding the k-periodic points of rational maps of ${\rm \mathbb{C}P}^1$

Maps, $f$, are understood in terms of a discrete steepest descent method, involving, as Lyapunov function, the log-chordal energy function associated to the fixed-points of $f$; ie the spherical Green's function rather than Coulomb energy. A transformation of rational maps of ${\rm \mathbb{C}P}^1$ which gives singular flat affine connections on ${\rm \mathbb{C}P}^1$ (also known as local systems, a complexification of polyhedra) in a natural way, provides an O(N,1)-intrinsic analogue of the Lyapunov force-fields and suggests higher dimensional versions of Schwartz-Christoffel uniformization of polygonal regions.

Relations to the algebraic geometry of configuration and moduli spaces, discriminants and dual curves are touched on, and we begin a discussion of the relation to geometric plethysm--maps as $sl_{2}\mathbb{C}$--invariants or covariants. We note as well the connection to moment maps, and begin a study of the relation of these constructions to hyperbolic centers of mass (such as Douady-Earle).

A class of self maps O(N,1)--intrinsic for hyperbolic space is constructed in each dimension as restrictions of the spherical rational maps above with fixed-points parameters in a hemisphere, generalizing the class of holomorphic maps of the 2-dimensional disc, and an associated "Schwarz lemma" confirms that the maps have good geometric and topological properties.

Received:
Jul 30, 2003
Published:
Jul 30, 2003
MSC Codes:
32, 53, 58
Keywords:
rational maps, equidistribution, schwarz lemma, equivariant

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Preprint
2003 Repository Open Access
Sidney Frankel

Equivariant rational maps and configurations : spherical equidistribution and SO(N,1) contraction