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Dynamics of McMullen maps

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Publication Date
Volume
229
Issue
4
Identifiers
DOI: 10.1016/j.aim.2011.12.026
Keywords
  • Local Connectivity
  • Yoccoz Puzzle
  • Mcmullen Map

Abstract

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.

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