Real Analyticity of Hausdorff Dimension for Expanding Rational Semigroups

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic families of semigroups satisfying all the above conditions and we analyze in detail the corresponding Bowen’s parameters and Hausdorff dimension function. Mathematics Subject Classification (2001). Primary 37F35; Secondary 37F15.