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 $U.$ 
First, we
classify these domains into three types (hyperbolic, simply parabolic
and doubly parabolic) according to the surface they induce when we quotient by the dynamics. 
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 $f$.