Alexandra Kaffl

• We investigate two combinatorial models for polynomial dynamics: Hubbard trees and kneading sequences. We consider abstract versions of unicritical and cubic Hubbard trees and kneading sequences and study their interdependence. This gives structure to the set of Hubbard trees, from which we derive results on the structure of the unicritical and cubic connectedness loci. In the first part, we consider unicritical Hubbard trees and kneading sequences and show that they are equivalent concepts. We determine which kneading sequences come from unicritical polynomials. On the parameter level we introduce a partial order < on the set of kneading sequences which is based on comparing periodic orbits in Hubbard trees. Transitivity of < follows from a forcing relation on periodic orbits between comparable Hubbard trees. Our main result in parameter space shows that the space of kneading sequences is structured like a tree. The motor of this study are results on the structure of periodic and precritical points in Hubbard trees. We conclude the unicritical part by discussing alternative approaches to define a partial order for the set of Hubbard trees. In the e second part, we investigate the interaction of abstract cubic Hubbard trees and kneading sequences. As a consequence of the existence of two critical points, it is not immediate how these concepts relate. Still, we show that there is an injection, but not a bijection, from the set of Hubbard trees into the set of kneading sequences. As in the unicritical case, we first explore the dynamical properties of Hubbard trees. On the parameter level, we define a partial order < on two special subsets of Hubbard trees, namely trees with only one free critical point and trees of disjoint type. Again we need a forcing relation on periodic orbits to establish transitivity of <. We give sufficient conditions that guarantee orbit forcing and provide examples that without them, orbit forcing might or might not be possible.