A Converse Landing Theorem in Parameter Spaces

TL;DR

This paper proves that in certain holomorphic families, parameter curves land at parabolic points, advancing understanding of ray structures in parameter spaces of transcendental entire maps.

Contribution

It establishes a general landing theorem for parameter curves in holomorphic families, extending known results to transcendental entire maps.

Findings

Existence of landing curves at parabolic parameters in specific families.

Partial answers to ray structure landing questions for transcendental maps.

Extension of classical landing results beyond quadratic and exponential families.

Abstract

In this article, we prove that for several one-dimensional holomorphic families of holomorphic maps, in the parameter plane, there exists a local piece of a curve that lands at a given parabolic parameter, in the spirit of well-known results about the quadratic and the exponential families. We also show that, under some assumptions, this general result partially answers the existence and landing questions of ray structures in the parameter planes for holomorphic families of transcendental entire maps.