Semi-hyperbolic fibered rational maps and rational semigroups

Abstract

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, we show that the fiberwise Julia sets are C₁-uniformly perfect and that the Hausdorff dimensions are greater than a positive constant C₂, where the constants C₁ and C₂ do not depend on any points in the base space. From this result we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere of degree two or greater, there exists a positive constant C such that the Julia set of any subsemigroup of G is C-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 k-porous, and the dynamics is weakly rigid. Moreover, we show that if fiberwise maps are polynomials, the fiberwise basins of infinity are c-John domains. For the above assertions, the constants k and c do not depend on any points in the base space. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on the Riemann sphere) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is porous or equal to the closure of the open set. Furthermore we derive an upper estimate of the Hausdorff dimension of the Julia set.

(Note: The author had to remove the section of entropy, which was in the previous version, because of the length of the paper. That section will be included in another paper. )

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