Summer Conference on Topology and Its Applications

Document Type

Semi-plenary Lecture

Publication Date


Publication Source

32nd Summer Conference on Topology and Its Applications


Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if we just fix the degree and impose the condition that only one critical orbit is finite. In that case, the family of rational maps defined by the combinatorics is a complex manifold naturally acted on by subgroups of the pure spherical braid group on n-strands where n depends on the order of the orbit and the degree, In this talk, we discuss the question: what is the global topology of this manifold?


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