Semihyperbolic transcendental semigroups

Kriete, Hartje; Sumi, Hiroki

doi:10.1215/kjm/1250517712

Cited by 4 publications

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“…For dynamics of semi-hyperbolic transcendental semigroups, see [KS1,KS2]. In this paper, by applying some of the results in [S4] obtained by the author, we show that semi-hyperbolicity along the fibers of a fibered rational map implies that the fiberwise Julia sets are k-porous, where the constant k does not depend on any points of the base space (hence the upper box dimensions of the fiberwise Julia sets are uniformly less than a constant that is strictly less than 2) (Theorem 1.16).…”

Section: Introductionmentioning

confidence: 99%

Semi-hyperbolic fibered rational maps and rational semigroups

Sumi

2006

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385705000532How to cite this article: HIROKI SUMI (2006). Semi-hyperbolic bered rational maps and rational semigroups. Ergodic Theory and Dynamical Systems, 26, Abstract. This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563-603). 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 the Riemann sphere 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 semihyperbolic, 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.

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Section: Introductionmentioning

confidence: 99%

Semi-hyperbolic fibered rational maps and rational semigroups

Sumi

2006

Ergod. Th. Dynam. Sys.

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“…The condition given in the following theorem is one of them. It was also obtained by Kriete and Sumi [14] in the case of semihyperbolic transcendental semigroups. For U ⊂ C we denote by diam(U ) the spherical diameter of U .…”

Section: Resultsmentioning

confidence: 70%

“…In this paper, we consider semihyperbolic entire functions. We note that Kriete and Sumi [14] treated the more general case of semihyperbolic entire semigroups, but our results are in a somewhat different direction. We show that semihyperbolic entire functions do not have wandering domains in which the iterates have a finite limit function.…”

Section: Introductionmentioning

confidence: 78%

Semihyperbolic entire functions

Bergweiler¹,

Morosawa²

2002

Nonlinearity

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“…Also, Ren et al studied such rational semigroups from the perspective of random dynamical systems [13,30]. Later, Kriete and Sumi [16] studied semigroups of entire maps.…”

Section: Introductionmentioning

confidence: 99%

Density of repelling fixed points in the Julia set of a rational or entire semigroup

Stankewitz

2010

Journal of Difference Equations and Applications

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Dedicated to Robert Devaney on the occasion of his 60th birthday We briefly survey several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, in particular, focusing on those methods which can be extended to the case of semigroups. We then present an elementary proof that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points.

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Semihyperbolic transcendental semigroups

Cited by 4 publications

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Semi-hyperbolic fibered rational maps and rational semigroups

Semi-hyperbolic fibered rational maps and rational semigroups

Semihyperbolic entire functions

Density of repelling fixed points in the Julia set of a rational or entire semigroup

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