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On the ergodicity of conformal measures for rational maps with totally disconnected Julia sets

Author: Yu Zhai
Journal: Proc. Amer. Math. Soc. 140 (2012), 3453-3462
MSC (2010): Primary 37F10, 37F20; Secondary 28D99
Published electronically: February 16, 2012
MathSciNet review: 2929014
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Abstract: Let $ f$ be a non-hyperbolic rational map with totally disconnected Julia set whose Fatou set is an attracting domain. In this paper, we prove that the number of ergodic components of any conformal measure for $ f$ is bounded by the number of critical points in its Julia set.

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Additional Information

Yu Zhai
Affiliation: Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing 100083, People’s Republic of China

Keywords: Conformal measure, ergodic component, Branner-Hubbard puzzle, KSS nest
Received by editor(s): October 22, 2010
Received by editor(s) in revised form: April 6, 2011
Published electronically: February 16, 2012
Additional Notes: The author was supported in part by China Postdoctoral Science Foundation Grant #20080440543.
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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