Baker domains and non-convergent deformations

Abstract

For an entire transcendental function $f$ with a non-completely invariant Baker domain $U$, we study the pinching process of paths in $U$ with certain restrictions, that we call Baker laminations. We show that if some curve in the Baker lamination of $f$ joins a point in the boundary of $U$ with infinity, then the deformation does not converge. Thus, in this particular case, the boundary of the space of deformations of $f$ is incomplete.