Mikhail Hlushchanka

• The past three decades have shown that lots of questions in holomorphic dynamics can be reduced to tractable combinatorial problems. One of the main objectives of this thesis is to gain better understanding of dynamics of the iteration of rational maps via developing good combinatorial models. The first such models, given by finite invariant graphs, were constructed for postcritically-finite polynomial maps and were used to classify these maps. However, the case of general rational maps is much more complicated and still draws lots of attention. In this work we construct combinatorial models for the family of expanding Thurston maps, which include all postcritically-finite rational maps with Julia sets given by the entire Riemann sphere. We show that each sufficiently large iterate of an expanding Thurston map has an invariant planar embedded tree containing the postcritical set. The latter result can be extended to the case of postcritically-finite rational maps with Sierpinski carpet Julia sets. In the thesis, we also provide a complete classification of critically fixed rational maps. The main tool is, again, a certain planar embedded invariant graph, called the Tischler graph, associated to each such map. We show that these graphs are always connected, answering a question raised by Pilgrim. We use the combinatorial models given by invariant graphs to study properties of the iterated monodromy groups (IMG's) of different classes of Thurston maps. In particular, we show that, in the presence of an invariant tree, the IMG's can be described in a very simple combinatorial way. This allows us to describe the IMG's that arise from critically fixed rational maps and conclude that these maps have amenable IMG's of exponential growth. Finally, in a joint work with Daniel Meyer, we construct conceptually new examples of rational maps with the IMG's of exponential growth.