@PhdThesis{diss_mods_00017449,
  author = 	{Albrecht, Simon},
  title = 	{On Functions in the Speiser Class with One Tract},
  year = 	{2015},
  publisher = 	{Christian-Albrechts-Universit{\"a}t zu Kiel},
  address = 	{Kiel},
  keywords = 	{Speiser class; Eremenko-Lyubich class; quasiconformal; quasiconformal folding; Bishop; quasiregular; tract; Speiser-Klasse; Eremenko-Lyubich-Klasse; quasikonform; quasikonforme Faltung; quasiregul{\"a}r; Trakt},
  abstract = 	{Let f be a transcendental entire map. A complex number w is called critical value of f if there exists a complex number z such that f'(z)=0 and f(z)=w. A complex number b is called an asymptotic value of f  if there exists a curve {\backslash}gamma with {\backslash}gamma(t){\backslash}to{\backslash}infty as t{\backslash}to{\backslash}infty but f({\backslash}gamma(t)){\backslash}to b as t{\backslash}to{\backslash}infty. The singular set of f is the set consisting of all critical and asymptotic values of f. The set B of all transcendental entire functions with a bounded singular set is called Eremenko-Lyubich class. The Speiser class S consists of all functions in class B where the singular set is not only bounded but finite.

These classes have been thoroughly studied, notably, in complex dynamics, which deals with the behaviour of an entire or rational map f under iteration. Of particular interest is the construction of functions in classes B and S with prescribed behaviour. One method to obtain maps in class B is using so-called Cauchy integrals. Gwyneth Stallard used this method to prove that for any d{\backslash}in(1,2) there exists a function in class B whose Julia set has Hausdorff dimension equal to d. It is the shape of the tracts of her maps which yields the desired Hausdorff dimension. Here, a tract is a connected component of the set where the modulus of the function is large.

The question arises whether Stallard's result also holds for maps in class S. While we are not able to answer this question, we show that there exist functions in class S whose tracts are in some sense similar to the tracts used by Stallard. The method of Cauchy integrals does generally not generate maps in class S. An alternative construction method is quasiconformal folding, which was recently introduced by Christopher Bishop.

We use Bishop's method to construct quasiregular maps which only grow in one parabola shaped tract which is symmetric to the real axis and are bounded otherwise. Furthermore, we prove that for each constructed quasiregular map g there exists an entire function f in class S such that g=f{\backslash}circ{\backslash}phi for some quasiconformal homeomorphism {\backslash}phi. Thus, the tract of f, which is still symmetric to the real axis, is the quasiconformal image of the tract of g. Moreover, the quasiconformal map involved is asymptotically conformal at infinity. We use this to prove that the maximum modulus M(r,f) of f on the circle with radius r is bounded below by a function which depends on the shape of the tract. In particular, we prove that there exists an entire map f in the class S with only one tract, which is symmetric to the real axis, such that {\backslash}log{\backslash}log M(r,f) is bounded below by d{\backslash}cdot{\backslash}sqrt{\{}r{\}} for some d>0.},
  url = 	{https://macau.uni-kiel.de/receive/diss_mods_00017449},
  file = 	{:https://macau.uni-kiel.de/servlets/MCRFileNodeServlet/dissertation_derivate_00006216/dissertation_simon_albrecht.pdf:PDF},
  language = 	{en}
}