On the connectivity of the Julia sets of meromorphic functions

TL;DR

This paper investigates the connectivity properties of Julia sets for meromorphic functions, proving new results about fixed points and domain structures, with implications for Newton's method and Baker domains.

Contribution

It establishes that transcendental meromorphic maps with disconnected Julia sets have weakly repelling fixed points and shows that Baker domains of Newton's method are simply connected.

Findings

Disconnected Julia sets imply the existence of weakly repelling fixed points.

Periodic Baker domains of Newton's method are proven to be simply connected.

Results extend to holomorphic self-maps of hyperbolic regions, showing the existence of absorbing domains.

Abstract

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.