Convergence of Rays with Rational Argument in Hyperbolic Components

TL;DR

This paper proves a theorem about the landing of rays with rational arguments in hyperbolic components using Carathéodory Convergence Theory, applicable to various families of entire transcendental maps.

Contribution

It introduces a new proof technique for ray landing theorems in hyperbolic components, generalizable to other one-parameter families of entire maps.

Findings

Proves a landing theorem for rays with rational arguments.

Uses Carathéodory Convergence Theory in the proof.

Method applicable to broader classes of entire transcendental maps.

Abstract

In this paper, we use the Carath\'eodory Convergence Theory to prove a landing theorem of rays in hyperbolic components with rational arguments. Although the proof is done in the setting of a family of entire transcendental maps with two singular values, the method can be generalized to many other one parameter families, with some modifications.