Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups
arXiv:0811.1809
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 equal to the Hausdorff dimension of the Julia set. Both -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 -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.
Published in Discrete and Continuous Dynamical Systems Ser. A., Vol 30, No. 1, 2011, 313--363. 50 pages, 2 figures