# American Institute of Mathematical Sciences

July  2011, 29(3): 929-952. doi: 10.3934/dcds.2011.29.929

## On the topology of wandering Julia components

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

Received  December 2009 Revised  May 2010 Published  November 2010

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.
Citation: Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
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