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Geometric measures for parabolic rational maps

Published online by Cambridge University Press:  19 September 2008

M. Denker  and
M. Urbański
M. Denker
Affiliation:
Institut für Mathematische Stochastik, Lotzestr. 13, 3400 Göttingen, Germany
M. Urbański
Affiliation:
Instytut Matematyki UMK, ul. Chopina 12/18, 87–100 Toruń, Poland
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Abstract

Let h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 1 , March 1992 , pp. 53 - 66
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[1]Aaronson, J., Denker, M. & Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. to appear.Google Scholar
[2]Blanchard, P.. Complex analytic dynamics of the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[3]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[4]Denker, M. & Urbański, M.. HausdorS and conformal measures on Julia sets with a rationally indifferent periodic point. J. London Math. Soc. 43 (1991), 107–118.Google Scholar
[5]Denker, M. & Urbański, M.. Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561–579.Google Scholar
[6]Guzmán, M.. Differentiation of integrals in ℝn. Springer Lecture Notes in Mathematics 481, Springer Verlag, 1975.Google Scholar
[7]Hille, E.. Analytic function theory. Ginn and Company: Boston-New York-Chicago-Atlanta-Dallas-Palo Alto-Toronto, 1962.Google Scholar
[8]Milnor, J.. Dynamics in one complex variable: Introductory Lectures, Preprint (1990).Google Scholar
[9]Sullivan, D.. Conformal dynamical systems. In: Geometric Dynamics. Springer Lecture Notes in Mathematics 1007, Springer Verlag, 1983, pp 725–752.CrossRefGoogle Scholar
[10]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.Google Scholar