Sarah Koch
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MATHEMATICS COLLOQUIUM
Speaker: Sarah Koch Abstract. In the 1980s, William Thurston, established his topological characterization of rational maps, one of the central results in the field of holomorphic dynamics. Thurston's theorem applies to postcritically finite rational maps (a rational map is postcritically finite if the orbit of every critical point eventually lands in a periodic cycle). Given such a rational map, one can define a holomorphic endomorphism of an associated complex manifold. This endomorphism has a unique fixed point, and the eigenvalues of the derivative at this fixed point are all algebraic. What do these eigenvalues mean? Do they have any geometric significance in the moduli space of rational maps? In the dynamical plane of the map itself? What algebraic numbers arise this way? We establish some facts about these eigenvalues, and we prove there are no "small eigenvalues" in the case of quadratic polynomials. The general situation is still quite mysterious. |