On Zorich maps and other topics in quasiregular dynamics

The work in this thesis revolves around the study of dynamical systems arising from iterating quasiregular maps. Quasiregular maps are a natural generalization of holomorphic maps in higher (real) dimensions and their dynamics have only recently started being systematically studied. We first study permutable quasiregular maps, i.e. maps that satisfy f ◦g = g ◦f, where we show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and g = φ ◦ f, where φ is a quasiconformal map, have the same Julia sets. Those results generalize well known theorems of Bergweiler, Hinkkanen and Baker on permutable entire functions. Next we study the dynamics of Zorich maps which are among the most important examples of quasiregular maps and can be thought of as analogues of the exponential map on the plane. For the exponential family Eκ : z 7→ κez, κ > 0, it has been shown that when κ > 1/e the Julia set of Eκ is the entire complex plane, essentially by Misiurewicz. Moreover, when n 0 < κ ≤ 1/e Devaney and Krych have shown that the Julia set of Eκ is an uncountable collection of disjoint curves. Bergweiler and Nicks have shown that a similar result is also true for Zorich maps. First we construct a certain "symmetric" family of Zorich maps, and we show that the Julia set of a Zorich map in this family is the whole of R3 when the value of the parameter is large enough, thus generalizing Misiurewicz's result. Moreover, we show that the periodic points of those maps are dense in R3 and that their escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential. On a similar note, we study the set of endpoints of the Julia sets of Zorich maps in the case that the Julia set is a collection of curves. We show that ∞ is an explosion point for the set of endpoints by introducing a topological model for the Julia sets of certain Zorich maps, similar to the so called straight brush of Aarts and Oversteegen. Moreover we introduce an object called a hairy surface which is a compactified version of the Julia set of Zorich maps and we show that those objects are not uniquely embedded in R3, unlike the corresponding two-dimensional objects which are all ambiently homeomorphic. Finally, we study the question of how a connected component of the inverse image of a domain under a quasiregular map covers the domain. We prove that the subset of the domain that is not covered can be at most of conformal capacity zero. This partially generalizes a result due to Heins. We also show that all points in this omitted set are asymptotic values