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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Attractors for graph critical rational maps


Authors: Alexander Blokh and Michal Misiurewicz
Journal: Trans. Amer. Math. Soc. 354 (2002), 3639-3661
MSC (2000): Primary 37F10; Secondary 37E25
DOI: https://doi.org/10.1090/S0002-9947-02-02999-9
Published electronically: April 30, 2002
MathSciNet review: 1911515
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Abstract: We call a rational map $f$ graph critical if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$.


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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email: ablokh@math.uab.edu

Michal Misiurewicz
Affiliation: Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: mmisiure@math.iupui.edu

DOI: https://doi.org/10.1090/S0002-9947-02-02999-9
Keywords: Complex dynamics, attractors, conformal measures, postcritical set
Received by editor(s): July 7, 2000
Received by editor(s) in revised form: December 20, 2001
Published electronically: April 30, 2002
Additional Notes: The first author was partially supported by NSF grant DMS 9970363
The second author was partially supported by NSF grant DMS 9970543
Article copyright: © Copyright 2002 American Mathematical Society