Univalent Baker domains and boundary of deformations

For $f$ an entire transcendental map with a univalent Baker domain $U$ of hyperbolic type I, we study pinching deformations in $U$, the support of this deformation being certain laminations in the grand orbit of $U$. We show that pinching along a lamination that contains the geodesic $λ_{\infty}$ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such $λ_{\infty}$, 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 $\infty$. 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 $M_{f}$ the marked space of topologically equivalent maps to $f$.

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