Dynamics of McMullen maps

Details Dynamics of McMullen maps Dynamics of McMullen maps

2012-04-09

arxiv.org/abs/1204.1907v1

Advances in Mathematics. 229(2012), 2525-2577

Mathematics

Dynamical Systems

Complex dynamics, 51 pages, 13 figures

Scientific paper

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.

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