No abstract

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes -the ivy graph.

The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and is coherent with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.