Mathematics – Dynamical Systems
Scientific paper
2008-11-12
Discrete and Continuous Dynamical Systems Ser. A., Vol 30, No. 1, 2011, 313--363
Mathematics
Dynamical Systems
Published in Discrete and Continuous Dynamical Systems Ser. A., Vol 30, No. 1, 2011, 313--363. 50 pages, 2 figures
Scientific paper
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.
Sumi Hiroki
Urbanski Mariusz
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