Let G be the Green function for a domain D $\subset$ R^d with d ≥ 3. The Martin boundary of D and the 3G inequality: $$\frac{G(x,y)G(y,z)}{G(x,z)} \le A(|x-y|^{2-d}+|y-z|^{2-d})\quad \text{for}\quad x,y,z \in D$$ 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.

We study the universal covering space $\tilde M$ of a holomorphic family (M, π, R) of Riemann surfaces over a Riemann surface R. The main result is that (1) $\tilde M$ is topologically equivalent to a two-dimensional cell, (2) $\tilde M$ is analytically equivalent to a bounded domain in C², (3) $\tilde M$ is not analytically equivalent to the two-dimensional unit ball B₂ under a certain condition, and (4) $\tilde M$ is analytically equivalent to the two-dimensional polydisc Δ² if and only if the homotopic monodoromy group of (M, π, R) is finite.

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.

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.

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 $\aleph_0$ of countably infinite set, and the cardinal number $\aleph$ of continuum, i.e. $\nabla\supset\mathbf{N}\cup\{\aleph_0,\aleph\}$, so that $\nabla=[1,\aleph]$, the interval of cardinal numbers ξ with $1\leq\xi\leq\aleph$, when the continuum hypothesis is postulated.

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.

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.

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).

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, $D^\vee=\{w\in\mathbf{C}:\, 1/w\in\hat{\mathbf{C}} \setminus\bar D\}$. This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.

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.

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