Measures and dimensions of Julia sets of semi-hyperbolicrational semigroups

Hiroki Sumi; Mariusz Urbański

doi:10.3934/dcds.2011.30.313

Article Contents

2011, Volume 30, Issue 1: 313-363. Doi: 10.3934/dcds.2011.30.313

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Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

Hiroki Sumi^1, and
Mariusz Urbański^2,

1.
Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043
2.
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received: August 2009

Revised: October 2010

Published: January 2011

Abstract / Introduction Related Papers Cited by

Abstract

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.

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Mathematics Subject Classification: Primary: 37F35; Secondary: 37F15.

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