Univalent Baker domains and boundary of deformations

TL;DR

This paper investigates how pinching deformations in univalent Baker domains of transcendental entire maps behave, revealing conditions under which these deformations converge to maps with wandering domains and their relation to Teichmüller space.

Contribution

It demonstrates convergence criteria for pinching deformations in univalent Baker domains and characterizes the resulting maps with wandering domains, linking to Teichmüller space structure.

Findings

Pinching along certain laminations does not converge.

Pinching along other laminations converges to a map with a wandering domain.

Conditions on Julia set and postcritical set influence convergence and limit maps.

Abstract

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 $\lambda_{\infty}$ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such $\lambda_{\infty}$, 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 Teichm\"uller space of $f$, $Teich(f)$, included in $M_{f}$ the marked space of topologically equivalent maps to $f$.