On the topology of wandering Julia components

Guizhen Cui; Wenjuan Peng; Lei Tan

doi:10.3934/dcds.2011.29.929

Article Contents

2011, Volume 29, Issue 3: 929-952. Doi: 10.3934/dcds.2011.29.929

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On the topology of wandering Julia components

1.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190
2.
Département de Mathématiques, Université d'Angers, Angers, 49045

Received: November 30, 2009

Revised: April 30, 2010

Published: August 2011

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Abstract

It is known that for a rational map $f$ with a disconnected Julia set, the set of wandering Julia components is uncountable. We prove that all but countably many of them have a simple topology, namely having one or two complementary components. We show that the remaining countable subset $\Sigma$ is backward invariant. Conjecturally $\Sigma$ does not contain an infinite orbit. We give a very strong necessary condition for $\Sigma$ to contain an infinite orbit, thus proving the conjecture for many different cases. We provide also two sufficient conditions for a Julia component to be a point. Finally we construct several examples describing different topological structures of Julia components.

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Mathematics Subject Classification: Primary: 37F10, 37F20.

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