Dynamics of McMullen maps

TL;DR

This paper studies the boundary and Julia set properties of McMullen maps using the Yoccoz puzzle technique, establishing conditions for connectedness, regularity, and local connectivity of the Julia set.

Contribution

It introduces the application of Yoccoz puzzle techniques to McMullen maps, proving boundary regularity and local connectivity results.

Findings

Boundary of the immediate basin of infinity is a Jordan curve if connected.

Boundary is a quasi-circle under certain conditions.

Julia sets are locally connected except in special cases.

Abstract

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to a question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.