Univalent Baker Domains and Boundary of Deformations

For f an entire transcendental map with univalent Baker domain U of hyperbolic type I, we study pinching deformations in U , the support of this deformation being a lamination in the grand orbit of U . We show that pinching along a lamination that contains the geodesic λ∞ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such λ∞, converges and converges to a unique map F if: the Julia set of f , J(f) is connected, the postcritical set of f is a positive (plane) distance away from J(f), and it is thin at ∞. We show that F has a simply connected wandering domain that stays away from the postcritical set. We interpret these results in terms of the Teichmüller space of f , Teich(f), included in Mf the marked space of topologically equivalent maps to f .