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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)) = ∞.

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

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