Real Analyticity of Hausdorff Dimension for Expanding Rational Semigroups
نویسندگان
چکیده
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.
منابع مشابه
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 generator...
متن کاملun 2 00 4 The dimensions of Julia sets of expanding rational semigroups ∗
We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.
متن کاملSe p 20 04 The dimensions of Julia sets of expanding rational semigroups ∗
We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.
متن کاملun 2 00 5 Dimensions of Julia sets of expanding rational semigroups ∗
We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.
متن کاملThe Dimensions of Julia Sets of Expanding Rational Semigroups *
We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.
متن کامل