On Hausdorff dimension of Julia sets of hyperbolic rational semigroups

Abstract

We will show that if a semigroup of rational functions on the Riemann sphere is finitely generated, then the hyperbolicity and the expandingness are equivalent. Also we consider finitely generated rational semigroups satisfying the strong open set condition. We show that if a semigroup satisfies the strong open set condition, we can construct a ^-conformal measure on the Julia set. Also the Julia set has no interior points, and furthurmore, if the semigroup is hyperbolic, the Hausdorff dimension of the Julia set is strictly lower than 2. The value δ of the dimension coincides with the unique value that allows us to construct a <5-conformal measure and the <5-Hausdorff measure of the Julia set is a finite value strictly bigger than zero. With the method similar to that of the construction of the Patterson-Sullivan measures we get (5-subconformal measures in more general cases and we will show that if a finitely generated rational semigroup is expanding, then the HausdorfF dimension of the Julia set is less than the exponent δ.