Spaces of Rational Maps: Dynamics, combinatorcs and topology

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The dynamics of complex rational maps is both very rich, and accessible to study. For this reason, they have been thoroughly investigated since the last 100 years, and particularly so recently, thanks to computer experimentation.The purpose of this proposal is to examine the space of rational maps, and exhibit its rich algebraic and combinatorial structure by combining tools from complex dynamics and group theory. We expect the language of “iterated monodromy groups” to be fundamental in building accurate combinatorial models of various dynamically-significant loci in the space of rational maps.An aside of this philosophy is that the space of rational maps should be understood via “post-critically finite maps”, namely maps such that the forward iterates of critical points form a finite set. Theseshould be thought of as akin to rational numbers among the reals dense, yet amenable to calculation.In particular, we will define and consider “slices” within the space of rational maps: One-dimensional subvarieties controlled by a finite collection of post-critically finite maps. The structure of the space of rational maps will then appear through the arrangement of these slices among themselves.