The Hornich topology for meromorphic functions in the disk

Pacific Journal of Mathematics, 1987

It is known that for 0 < p < oo the Hardy space H p contains a residual set of functions, each of which has range equal to the whole plane at every boundary point of the unit disk. With quite new general techniques, we are able to show that this result holds for numerous other spaces. The space BMOA of analytic functions of bounded mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet spaces D a f or 0 < a < 1/2 are examples. Our methods involve hyperbolic geometry, cluster set analysis and the "depth" function which we have used previously for determining geometric properties of the image surfaces of functions. Denote by D(a, r) the open disc in C centered at a and of radius r. Denote by D the unit disc D(0,l) and let Δ(a,r) = D Π D{a,r) for a e 3D. Brown and Hansen [4] proved that each Hardy space H p 9

Eprint Arxiv Math 9905127, 1999

Germs of meromorphic functions has recently become an object of study in singularity theory. T. Suwa ([11]) described versal deformations of meromorphic germs. V.I. Arnold ([1]) classified meromorphic germs with respect to certain equivalence relations. The authors ([4]) started a study of topological properties of meromorphic germs. Some applications of the technique developed in [4] were described in [5] and [6]. In [4] the authors elaborated notions and technique which could be applied to compute such invariants of polynomials as Euler characteristics of fibres and zetafunctions of monodromy transformations associated with a polynomial (see [5]). Some crucial basic properties of the notions related to the topology of meromorphic germs were not discussed there. This has produced some lack of understanding of the general constructions. The aim of this note is to partially fill in this gap. At the same time we describe connections with some previous results and generalizations of them. A polynomial P in n + 1 complex variables defines a map P from the affine complex space C n+1 to the complex line C. It is well known that the map P is a C ∞-locally trivial fibration over the complement to a finite set in the line C. The smallest of such sets is called the bifurcation set or the set of atypical values of the polynomial P. One is interested in describing the topology of the fibre of this fibration and its behaviour under monodromy transformations corresponding to loops around atypical values of the polynomial P. The monodromy transformation corresponding to a circle of big radius which contains all atypical values (the monodromy transformation of the polynomial P at infinity) is of particular interest. The initial idea was to reduce calculation of the zeta-function of the monodromy transformation at infinity (and thus of the Euler characteristic of the generic fibre) of the polynomial P to local problems associated to different points at infinity, i.e., at the infinite hyperplane CP n ∞ in the projective compactification CP n+1 of the affine space C n+1. The possibility of such a localization for holomorphic germs was used in [3]. This localization can be expressed in terms of an integral with respect to the Euler characteristic, a notion introduced by the school of V.A. Rokhlin ([12]). However the results are not apply directly to a polynomial function since at a point of the infinite hyperplane CP n ∞ a polynomial function defines not a Key words and phrases. Germs of meromorphic functions, Milnor fibre, atypical values.

Bulletin of the American Mathematical Society, 1984

Bulletin of The American Mathematical Society, 1975

2008

Diverse structural properties for classes of holomorphic functions (defined by means of maximal functions) are proved, including the Riesz convergence theorem and a factorization theorem.

Pacific Journal of Mathematics, 1985

By a classical result of Fatou, a bounded analytic function on the unit disc £>, i.e. in the space H°°(D), has a radial limit at almost every point on diλ We examine the question of whether this limiting or boundary value lies in the interior or on the boundary of the image domain. We show that the first case is "typical" in the sense that every function in a certain dense G δ-set of H°° has this property at a.e. boundary point. Several other spaces including the disc algebra and the Dirichlet space are also studied.

Science in China Series A: Mathematics, 2005

In this paper we relate the study of unique range sets for meromorphic functions (URSM) with the hyperbolic hypersurfaces and give some remarks on the genericity of unique range sets for meromorphic functions.

Studia Mathematica, 2006

Let X be a Riemann domain over C k ×C l. If X is a domain of holomorphy with respect to a family F ⊂ O(X), then there exists a pluripolar set P ⊂ C k such that every slice X a of X with a / ∈ P is a region of holomorphy with respect to the family {f | X a : f ∈ F }. 1. Introduction: Riemann regions of holomorphy. Let (X, p) be a Riemann region over C n , i.e. X is an n-dimensional complex manifold and p : X → C n is a locally biholomorphic mapping (see [Jar-Pfl 2000] for details). If X is connected, then (X, p) is said to be a Riemann domain. We say that two Riemann regions (X, p) and (Y, q) over C n are isomorphic (written (X, p) ≃ (Y, q)) if there exists a biholomorphic mapping ϕ : X → Y such that q • ϕ = p. Throughout, isomorphic Riemann regions will be identified. We say that an open set U ⊂ X is univalent (schlicht) if p| U is injective. Note that X is univalent iff (X, p) ≃ (Ω, id Ω), where Ω is an open set in C n. Let f ∈ O(X). For any α ∈ Z n + (Z + stands for the set of non-negative integers) and x 0 ∈ X, let D α f (x 0) denote the α-partial derivative of f at x 0 , D α f (x 0) := D α (f • (p| U) −1)(p(x 0)), where U is an open univalent neighborhood of x 0 and D α on the right hand side means the standard α-partial derivative operator in C n. Let T x 0 f denote the Taylor series of f at x 0 , i.e. the formal power series α∈Z n + 1 α! D α f (x 0)(z − p(x 0)) α , z ∈ C n. For x 0 ∈ X and 0 < r ≤ ∞ let P X (x 0 , r) denote an open univalent neighborhood of x 0 such that p(P X (x 0 , r)) = P(p(x 0), r) = the polydisc

Proceedings of the Steklov Institute of Mathematics, 2017

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

1973