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Journal of the London Mathematical Society 1991 s2-43(1):107-118; doi:10.1112/jlms/s2-43.1.107
© 1991 by London Mathematical Society
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© Oxford University Press

Hausdorff and Conformal Measures on Julia Sets with a Rationally Indifferent Periodic Point

M. Denker and M. Urbanski

Institut für Mathematische Stochastik Lotzestraße 13, 3400 Göttingen, Germany
Instytut Matematyki, N. Copernicus University ul. Chopina 12/18, 87-100 Torun, Poland

We show that the Hausdorff dimension {delta} of a non-hyperbolic Julia set J(T) without critical points can be expressed by the smallest zero of the pressure function t↦P(T, – t log |T'|). This result is similar to the Bowen-Manning-McCluskey formula. The Hausdorff dimension is also shown to be the smallest exponent t{varepsilon}R for which a t-conformal measure in the sense of Sullivan exists. We also prove uniqueness properties of t-conformal measures, and we prove the absolute continuity of the Hausdorff measure H{delta} with respect to any {delta}-conformal measure.


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