Se p 20 05 Semi - hyperbolic fibered rational maps and rational semigroups ∗
نویسنده
چکیده
This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on C of degree two or greater, the Julia set of any subsemigroup of G is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on C) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.
منابع مشابه
ar X iv : m at h / 05 09 71 9 v 3 [ m at h . D S ] 1 0 Ju n 20 06 Semi - hyperbolic fibered rational maps and rational semigroups ∗
This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a com...
متن کاملSemi-hyperbolic fibered rational maps and rational semigroups
This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a com...
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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.
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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.
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In 1980’s, Thurston established a combinatorial characterization for post-critically finite rational maps among post-critically finite branched coverings of the two sphere to itself. A completed proof was written by Douady and Hubbard in their paper [A. Douady, J.H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993) 263–297]. This criterion ...
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