Structural theorems for holomorphic self-maps of the punctured plane

Abstract

This thesis concerns the iteration of transcendental self-maps of the punctured plane C*:=C\{0}, that is, functions f : C*→C* that are holomorphic on C* and for which both zero and infinity are essential singularities. We focus on the escaping set of such functions, which consists of the points whose orbit accumulates to zero and/or infinity under iteration. The escaping set is closely related to the structure of the phase space due to its connection with the Julia set. We introduce the concept of essential itinerary of an escaping point, which is a sequence that describes how its orbit accumulates to the essential singularities, and plays a very important role throughout the thesis. This allows us to partition the escaping set into uncount-ably many non-empty subsets of points that escape in non-equivalent ways, the boundary of each of which is the Julia set. We combine the iterates of the maximum and minimum modulus functions to define the fast escaping set for functions in this class and, for such functions, construct orbits with several types of annular itinerary, including fast escaping and arbitrarily slowly escaping points. Next we proceed to study in detail the class B* of bounded-type transcendental self-maps of C*, for which the escaping set is a subset of the Julia set, so such functions do not have escaping Fatou components. We show that, for finite compositions of transcendental self-maps of C* of finite order (and hence in B*), every escaping point can be joined to one of the essential singularities by a curve of points that escape uniformly. Moreover, we prove that, for every essential itinerary, the corresponding escaping set contains a Cantor bouquet and, in particular, uncountably many such curves. Finally, in the last part of the thesis we direct our attention to the functions that do have escaping Fatou components. We give the first explicit examples of transcendental self-maps of C* with Baker domains and escaping wandering domains and use approximation theory to construct functions with escaping Fatou components that have any prescribed essential itinerary.