Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map f̃ and the Lyapunov exponent of f̃ with respect to the maximal entropy measure for f̃ . Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form ajz by a Möbius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than 2. Mathematics Subject Classification (2001). Primary 37F35; Secondary 37F15.