Kodai Mathematical Journal

The 3G inequality for a uniformly John domain

Hiroaki Aikawa, Torbjörn Lundh

2005 年 28 巻 2 号 p. 209-219
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767003

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Let G be the Green function for a domain D⊂R^d with d≥3. The Martin boundary of D and the 3G inequality:
\frac{G(x, y)G(y, z)}{G(x, z)}≤A(|x−y|^2−d+|y−z|^2−d) for x, y, z∈D
are studied. We give the 3G inequality for a bounded uniformly John domain D, although the Martin boundary of D need not coincide with the Euclidean boundary. On the other hand, we construct a bounded domain such that the Martin boundary coincides with the Euclidean boundary and yet the 3G inequality does not hold.

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A lower bound of maximal dilatations of quasiconformal automorphisms acting on Fuchsian groups

Ege Fujikawa

2005 年 28 巻 2 号 p. 220-229
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767004

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A remark on universal coverings of holomorphic families of Riemann surfaces

Yoichi Imayoshi, Minori Nishimura

2005 年 28 巻 2 号 p. 230-247
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767005

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We study the universal covering space ˜{M} of a holomorphic family (M, π, R) of Riemann surfaces over a Riemann surface R. The main result is that (1) ˜{M} is topologically equivalent to a two-dimensional cell, (2) ˜{M} is analytically equivalent to a bounded domain in C², (3) ˜{M} is not analytically equivalent to the two-dimensional unit ball C₂ under a certain condition, and (4) ˜{M} is analytically equivalent to the two-dimensional polydisc Δ² if and only if the homotopic monodoromy group of (M, π, R) is finite.

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Jørgensen groups of parabolic type III (uncountably infinite case)

Changjun Li, Makito Oichi, Hiroki Sato

2005 年 28 巻 2 号 p. 248-264
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767006

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Jørgensen groups of parabolic type parametrized by three real parameters are divided into three types: finite type, countably infinite type and uncoutably infinite type. In the previous papers we found all Jørgensen groups of finite type and of countably infinite type. In this paper we find all Jørgensen groups of uncoutably infinite type. Consequently, the problem finding all Jørgensen groups of these parabolic type has been completely solved.

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Conformal slit mappings from periodic domains

Fumio Maitani

2005 年 28 巻 2 号 p. 265-274
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767007

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Quasiconformal mappings and minimal Martin boundary of p-sheeted unlimited covering surfaces of the complex plane

Hiroaki Masaoka, Shigeo Segawa

2005 年 28 巻 2 号 p. 275-279
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767008

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Circularizable domains on Riemann surfaces

Makoto Masumoto, Masakazu Shiba

2005 年 28 巻 2 号 p. 280-291
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767009

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Isoperimetric constants for conservative Fuchsian groups

Katsuhiko Matsuzaki

2005 年 28 巻 2 号 p. 292-300
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767010

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The critical exponents of conservative Fuchsian groups are bounded from below by 1/2. It is proved in this note that this result is sharp by giving a sequence of conservative Fuchsian groups whose critical exponents converge to 1/2. The proof is carried out by estimating the isoperimetric constants of hyperbolic surfaces associated with the Fuchsian groups.

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The limit sets of quasifuchsian punctured surface groups and the Teichmüller distances

Hideki Miyachi

2005 年 28 巻 2 号 p. 301-309
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767011

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Heins problem on harmonic dimensions

Mitsuru Nakai, Toshimasa Tada

2005 年 28 巻 2 号 p. 310-327
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767012

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The main assertion of this paper is that for an arbitrarily given parabolic open Riemann surface R there always exists a Heins surface W_R, i.e. a parabolic open Riemann surface with the single ideal boundary component, such that the harmonic dimension of W_R, i.e. the cardinal number of the set of minimal Martin boundary points of W_R, is identical with that of R. The result is then applied to give a simple and unified proof for the best theorem at present as an answer to the Heins problem to determine the set ∇ of harmonic dimensions of all Heins surfaces obtained by collecting contributions of many authors that ∇ contains the set N of all positive integers, the cardinal number ℵ₀ of countably infinite set, and the cardinal number ℵ of continuum, i.e. ∇⊃N∪{ℵ₀, ℵ}, so that ∇=[1, ℵ], the interval of cardinal numbers ξ with 1≤ξ≤ℵ, when the continuum hypothesis is postulated.

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Plane domains with harmonic interpolating sequences which are not interpolating sequences

Junichiro Narita

2005 年 28 巻 2 号 p. 328-335
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767013

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A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka's Theorem

Junjiro Noguchi

2005 年 28 巻 2 号 p. 336-346
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767014

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Linearization problem on structurally finite entire functions

Yûsuke Okuyama

2005 年 28 巻 2 号 p. 347-358
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767015

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We show that if a 1-hyperbolic structurally finite entire function of type (p, q), p≥1, is linearizable at an irrationally indifferent fixed point, then its multiplier satisfies the Brjuno condition. We also prove the generalized Mañé theorem; if an entire function has only finitely many critical points and asymptotic values, then for every such a non-expanding forward invariant set that is either a Cremer cycle or the boundary of a cycle of Siegel disks, there exists an asymptotic value or a recurrent critical point such that the derived set of its forward orbit contains this invariant set. From it, the concept of n-subhyperbolicity naturally arises.

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Best approximation, Tikhonov regularization and reproducing kernels

Saburou Saitoh

2005 年 28 巻 2 号 p. 359-367
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767016

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We shall give a definite and good application of the theory of reproducing kernels to the Tikhonov regularization that is powerful in best approximation problems in numerical analysis.

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On complex analytic properties of limit sets and Julia sets

Hiroshige Shiga

2005 年 28 巻 2 号 p. 368-381
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767017

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It is known that Kleinian groups and complex dynamics have many properties in common. We shall show that the structure of the Martin boundaries of the region of discontinuity and the Fatou set are similar under a certain condition. We also show that any conformal mapping defined on the region of discontinuity (resp. the Fatou set) has a continuous extension on the conical limit set (resp. the conical Julia set in the sense of Lyubich and Minsky).

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A self-duality of strong starlikeness

Toshiyuki Sugawa

2005 年 28 巻 2 号 p. 382-389
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767018

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In this note, we will show that a simply connected bounded domain D is strongly starlike of order α<1 with respect to the origin if and only if so is D^∨, where D^∨ is the analytic inversion of the exterior of D, namely, D^∨={w∈C : 1/w∈\hat{C}{\backslash}\bar{D}}. This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.

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Dimensions of Julia sets of expanding rational semigroups

Hiroki Sumi

2005 年 28 巻 2 号 p. 390-422
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767019

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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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Holomorphic maps of Riemann surfaces and Weierstrass points

Masaharu Tanabe

2005 年 28 巻 2 号 p. 423-429
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767020

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We investigate holomorphic maps between hyperelliptic Riemann surfaces and prove rigidity theorems given by correspondences of Weierstrass points.

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Covering structure and dynamical structure of a structurally finite entire function

Masahiko Taniguchi

2005 年 28 巻 2 号 p. 430-438
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767021

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Word length and limit sets of Kleinian groups

Tsuyoshi Yamaguchi

2005 年 28 巻 2 号 p. 439-451
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767022

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Regions of variability for functions of bounded derivatives

Hiroshi Yanagihara

2005 年 28 巻 2 号 p. 452-462
発行日: 2005年
公開日: 2011/01/04

DOIhttps://doi.org/10.2996/kmj/1123767023

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