Recurrent critical points and typical limit sets of rational maps
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- by Alexander M. Blokh, John C. Mayer and Lex G. Oversteegen PDF
- Proc. Amer. Math. Soc. 127 (1999), 1215-1220 Request permission
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.References
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Additional Information
- Alexander M. Blokh
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 196866
- 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
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1215-1220
- MSC (1991): Primary 58F23, 28D99; Secondary 34C35
- DOI: https://doi.org/10.1090/S0002-9939-99-04721-8
- MathSciNet review: 1485461