Preprint 74/2003

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

Sidney Frankel

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Submission date: 30. Jul. 2003 (revised version: September 2003)
Pages: 63
MSC-Numbers: 32, 53, 58
Keywords and phrases: rational maps, equidistribution, schwarz lemma, equivariant
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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 formula62. We construct geometrically natural examples of rational maps of formula64, and introduce a new approach, ``suspension'', to producing iterative algorithms for factoring polynomials, and to finding the k-periodic points of rational maps of formula62

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 formula62 which gives singular flat affine connections on formula62 (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 formula76-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-point 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.

07.06.2018, 02:11