DEFORMATION OF ENTIRE FUNCTIONS WITH BAKER DOMAINS
Nuria Fagella (*), CHRISTIAN HENRIKSEN (**)
(*)
Departament de Matematica aplicada i Analisi
Universitat de Barcelona
Gran Via 585
08005 Barcelona
Spain
e-mail: fagella@maia.ub.es
(**)
Department of mathematics
Technical university of Denmark
Building 303
DK-2800 Lyngby, Denmark
e-mail: christian.henriksen@mat.dtu.dk
ABSTRACT:
We consider entire transcendental functions $f$ with an invariant (or
periodic) Baker domain $U$ satisfying a certain condition (which is satisfied always if $f$ restricted to $U$ is proper).
First, we
classify these domains into three types (hyperbolic, simply parabolic
and doubly parabolic) according to the properties
of the map they induce in the unit disk, and we give dynamical and
geometric criteria to determine the type of a given Baker domain.
Second, we study the space of quasiconformal
deformations of an entire map with such a Baker domain by studying
its Teichm\"uller space. More precisely, we show that the dimension of
this set is infinite if the Baker domain is hyperbolic or simply parabolic, and
from this we deduce that the quasiconformal deformation space of
$f$ is infinite dimensional. Finally, we prove that the function
$f(z)=z+e^{-z}$, which possesses infinitely many invariant Baker
domains, is rigid, i.e., any quasiconformal deformation of $f$ is
affinely conjugate to $f$.