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 of the Riemann sphere with normalized Lebesgue measure and show that if there is a subset of the Julia set of positive -measure whose points have limit sets not contained in the union of the limit sets of *recurrent* critical points, then for -a.e. point and is conservative, ergodic and exact.

**[Bar97]**J. A. Barnes,*Conservative exact rational maps of the sphere*, preprint (1997).**[CG93]**Lennart Carleson and Theodore W. Gamelin,*Complex dynamics*, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR**1230383****[Lyu83]**M. Yu. Lyubich,*Typical behavior of trajectories of the rational mapping of a sphere*, Dokl. Akad. Nauk SSSR**268**(1983), no. 1, 29–32 (Russian). MR**687919****[McM94]**Curtis T. McMullen,*Complex dynamics and renormalization*, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR**1312365****[Mil92]**J. Milnor,*Locally connected Julia sets: Expository lectures*, Tech. Report 11, SUNY-Stony Brook, 1992, Institute for Mathematical Sciences.**[Mn93]**Ricardo Mañé,*On a theorem of Fatou*, Bol. Soc. Brasil. Mat. (N.S.)**24**(1993), no. 1, 1–11. MR**1224298**, 10.1007/BF01231694**[Roh64]**V. Rohlin,*Exact endomorphisms of a Lebesgue space*, Amer. Math. Soc. Transl.**39(2)**(1964), 1-36.**[Urb94]**Mariusz Urbański,*Rational functions with no recurrent critical points*, Ergodic Theory Dynam. Systems**14**(1994), no. 2, 391–414. MR**1279476**, 10.1017/S0143385700007926

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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

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