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Rational functions with no recurrent critical points

Published online by Cambridge University Press:  19 September 2008

Mariusz Urbański
Mariusz Urbański
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA
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Abstract

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 391 - 414
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

[ADU]Aaronson, J., Denker, M. & Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.CrossRefGoogle Scholar
[BI]Blanchard, P.. Complex analytic dynamics of the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[Br]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[DU1]Denker, M. & Urbański, M.. Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point. J. London Math. Soc. 43 (1991), 107–118.CrossRefGoogle Scholar
[DU2]Denker, M. & Urbański, M.. On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561–579.CrossRefGoogle Scholar
[DU3]Denker, M. & Urbański, M.. On Sullivan's conformal measures for rational maps of the Riemann sphere. Nonlinearity 4 (1991), 365–384.CrossRefGoogle Scholar
[DU4]Denker, M. & Urbański, M.. On Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76 (1992), 193–214.CrossRefGoogle Scholar
[DU5]Denker, M. & Urbański, M.. Geometric measures for parabolic rational maps. Ergod. Th. and Dynam. Sys. 12 (1992), 53–66.CrossRefGoogle Scholar
[DU6]Denker, M. & Urbański, M.. The capacity of parabolic Julia sets. Math. Zeitsch. 211 (1992), 73–86.CrossRefGoogle Scholar
[F]Federer, H.F.. Geometric Measure Theory Springer: New York, 1969.Google Scholar
[G]Guzman, M.. Differentiation of integrals in ℝn. Springer Lecture Notes in Mathematics 481. Springer: New York, 1975.CrossRefGoogle Scholar
[H]Hille, E.. Analytic Function Theory. Ginn and Company: Boston-New York-Chicago-Atlanta-Dallas- Palo Alto-Toronto, 1962.Google Scholar
[Ma]Mañé, R.. On a theorem of Fatou. Preprint.Google Scholar
[Mi]Milnor, J.. Dynamics in one complex variable: Introductory lectures. Preprint. (1990).Google Scholar
[P]Przytycki, F.. Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119(1) (1993), 309–317.CrossRefGoogle Scholar
[S1]Sullivan, D.. Conformal dynamical systems. In: Geometric Dynamics. Springer Lecture Notes in Mathematics 1007. Springer: New York, 1983. pp. 725–752.CrossRefGoogle Scholar
[S2]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.CrossRefGoogle Scholar