Invariants and Parameter Space Models for Rational Maps

This thesis deals with classification of special classes of 1-variable holomorphic rational functions. In the first part we focus on the class of rational maps with bounded orbits of post-critical points – so-called Thurston maps. These maps can be viewed as a class of topological objects – branching coverings. There exists a natural equivalence relation on the class of Thurston maps, such that different rational functions are almost never equivalent. We provide an algorithm which allows to represent these algebraic (or topological) objects by means of completely combinatorial objects – invariant graphs with marked vertices. We also introduce a computational procedure for finding such graphs. Then we look at the cubic polynomials with connected Julia sets with a fixed multiplier, thus we get a slice of such polynomials. Such slices can be considered as parameter spaces. We introduce a parametrization of these cubic slices via reglued Julia set of the quadratic polynomial with the same multiplier. We also show that the parametrizing map is continuous. 2