Colloquium Series Seminar

A rational function f(z) with complex coefficients defines a holomorphic self-map of the Riemann sphere CP^1. The dynamics of f, namely, its behavior under iteration, are in many ways predicted by the orbits of critical points - points on CP^1 where the derivative of f is zero. A rational function f is called postcritically finite if the orbit of every critical point is finite. Topologically, a rational function is a branched covering, i.e. a covering map except over the images of critical points. W. Thurston characterized postcritically finite branched coverings of a sphere S^2 that are homotopic to postcritically finite rational functions on CP^1.

Let phi:S^2 -> S^2 be a postcritically finite branched covering with postcritical set P. The branched covering phi induces a holomorphic self-map T(phi) of the Teichmuller space of complex structures on (S^2, P). Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems. If phi meets certain combinatorial criteria, then the inverse of the transcendental map T(phi) descends to a rational/meromorphic self-map R(phi) of projective space P^N. If, in addition, phi has a fully ramified fixed point (i.e. phi is a "topological polynomial") , then R(phi) is regular/holomorphic.

I will discuss how the algebraic dynamics of R(phi) are influenced by the topological dynamics of phi. In particular, the more phi resembles a topological polynomial, the more the meromorphic map R(phi) resembles a holomorphic map. Speaker(s): Rohini Ramadas (Harvard University)