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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Recurrent critical points and typical
limit sets of rational maps


Authors: Alexander M. Blokh, John C. Mayer and Lex G. Oversteegen
Journal: Proc. Amer. Math. Soc. 127 (1999), 1215-1220
MSC (1991): Primary 58F23, 28D99; Secondary 34C35
MathSciNet review: 1485461
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of the Riemann sphere with normalized Lebesgue measure $\mu$ and show that if there is a subset of the Julia set $J(f)$ of positive $\mu$-measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then $\omega(x)=\widehat{\mathbb{C}}=J(f)$ for $\mu$-a.e. point $x$ and $f$ is conservative, ergodic and exact.


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Additional Information

Alexander M. Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: ablokh@math.uab.edu

John C. Mayer
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: mayer@math.uab.edu

Lex G. Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: overstee@math.uab.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04721-8
PII: S 0002-9939(99)04721-8
Keywords: Julia set, complex analytic dynamics, limit set, recurrent critical point
Received by editor(s): July 23, 1997
Additional Notes: The first author was partially supported by NSF Grant DMS 9626303 and the third author by NSF Grant DMS 9704903. Portions of this paper were presented by the third author at the Spring Topology Conference in Lafayette, Louisiana, in April, 1997
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society