Local connectivity of boundaries of tame Fatou components of meromorphic functions

TL;DR

This paper proves local connectivity of boundaries of invariant Fatou components for a broad class of transcendental meromorphic functions, including Newton's method for sine, even with infinitely many unbounded components.

Contribution

It establishes local connectivity results for Fatou boundaries without requiring geometric finiteness or class , covering cases with infinite post-singular values and essential singularities.

Findings

Boundaries of invariant Fatou components are locally connected.

Results apply to Newton's method for inf3 sine, with infinitely many unbounded components.

First example of a locally connected Julia set for a transcendental map outside class .

Abstract

We prove local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class $\mathcal B$, and the boundaries of the basins (possibly unbounded) are allowed to contain an infinite number of post-singular values, as well as the essential singularity at infinity. A basic assumption is that the unbounded parts of the basins are contained in regions which we call `repelling petals at infinity', where the map exhibits a kind of `parabolic' behaviour. In particular, our results apply to a wide class of Newton's methods for transcendental entire maps. As an application, we prove local connectivity of the Julia set of Newton's method for $\sin z$, providing the first non-trivial example of a locally connected Julia set of a transcendental map outside class $\mathcal B$, with an infinite number of unbounded Fatou components.