Structural Theorems for Holomorphic Self-Maps of the Punctured Plane

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

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class B∗, that is, transcendental self-maps of C∗ for which the set of singular values is contained in a compact annulus of C∗ that separates zero from infinity. For functions in the class B∗, escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of C∗ (and hence, in the class B∗), then we show that every escaping point of f can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence e∈{0,∞}N0, we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to the set {0,∞} according to e under iteration by f

On the connectivity of the escaping set in the punctured plane

We consider the dynamics of transcendental self-maps of the punctured plane, C∗=C∖{0}. We prove that the escaping set I(f) is either connected, or has infinitely many components. We also show that I(f)∪{0,∞} is either connected, or has exactly two components, one containing 0 and the other ∞. This gives a trichotomy regarding the connectivity of the sets I(f) and I(f)∪{0,∞}, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in C∗ by constructing the first such example. We also prove that if f has a doubly connected Baker domain, then its closure contains both 0 and ∞, and hence I(f)∪{0,∞} is connected

Bounded Fatou and Julia components of meromorphic functions

We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.Comment: 15 pages, 4 figures. V2: We have revised the introduction, and introduced two new sections: Section 2 discusses and compare topological properties of Fatou components, while Section 3 establishes that certain bounded regular domains cannot arise as eventually periodic Fatou components of meromorphic function

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class B∗, that is, transcendental self-maps of ℂ∗ for which the set of singular values is contained in a compact annulus of ℂ∗ that separates zero from infinity. For functions in the class B∗, escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of ℂ∗ (and hence, in the class B∗), then we show that every escaping point of f can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence e ∈ {0,∞}, we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to the set {0,∞} according to e under iteration by f