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

Abstract

We consider the dynamics of semi-hyperbolic semigroups generated byfinitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holdsit 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 Hausdorffand packing measures are finite and positive on the Julia set and are mutually equivalentwith Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractaldimensions, Hausdorff, packing and box counting are equal. It is also proved that forthe 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 absolutelycontinuous invariant (under the skew-product map) measure. In fact these two measures areequivalent, and the invariant measure is metrically exact, hence ergodic.