Title:

Structural theorems for holomorphic selfmaps of the punctured plane

This thesis concerns the iteration of transcendental selfmaps of the
punctured plane C* := C \ {O}, 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 uncountably
many nonempty subsets of points that escape in nonequivalent
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 13* of boundedtype
transcendental selfmaps of C*, for which the escaping set is a sub
set of the Julia set, so such functions do not have escaping Fatou
components. We show that, for finite compositions of transcendental
selfmaps 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 selfmaps 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.
This thesis concerns the iteration of transcendental selfmaps of the
punctured plane C* := C \ {O}, 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 uncountably many nonempty subsets of points that escape in nonequivalent
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 boundedtype
transcendental selfmaps of C*, for which the escaping set is a sub
set of the Julia set, so such functions do not have escaping Fatou
components. We show that, for finite compositions of transcendental selfmaps 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 selfmaps 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.
