Astérisque N° 288/2003 - Grand Format

In this work, we investigate the structure of certain parameter spaces. The aim is to understand the variation of dynamics - in particular, of hyperbolic dynamics - in certain parameter spaces of rational maps. In order to do this, we examine the topological and geometric structure of larger parameter spaces, of branched coverings of the Riemann sphere C, where some of the critical points are constrained to have finite forward orbits.
We obtain a complete topological description of the spaces under consideration, from two points of view, which we call the Topographer's View and the Resident's View. The Topographer's View is, in essence, a geometrising theorem. It shows that the space in question is, up to homotopy equivalence, a countable union of disjoint geometric pieces, joined together by handles. The most typical geometric pieces are varieties of rational maps, and tori.
The Resident's View is a view of the whole parameter space from the dynamical plane of a map (a resident) in the parameter space. This is necessarily a two-dimensional view, in which the geometric pieces of the parameter space appear as disjoint convex regions in the dynamical plane.