# American Institute of Mathematical Sciences

May  2006, 15(2): 379-394. doi: 10.3934/dcds.2006.15.379

## Deformation of entire functions with Baker domains

 1 Dept. de Mat. Aplicada i Anàlisi, Univ. de Barcelona, Gran Via 585, 08007 Barcelona, Spain 2 The Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark

Received  June 2004 Revised  December 2005 Published  March 2006

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 take the quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmüller 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 $f$.
Citation: Núria Fagella, Christian Henriksen. Deformation of entire functions with Baker domains. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 379-394. doi: 10.3934/dcds.2006.15.379
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