Elsevier

Advances in Mathematics

Volume 229, Issue 4, 1 March 2012, Pages 2525-2577
Advances in Mathematics

Dynamics of McMullen maps

https://doi.org/10.1016/j.aim.2011.12.026Get rights and content
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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 the 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.

MSC

37F10
37F45

Keywords

Local connectivity
Yoccoz puzzle
McMullen map

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