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 $f$.