Fibre cycles of holomorphic maps

On leave from: Mathematisches Institut, Bunsenstrasse 3-5, D-37073, Göttingen, Germany     

Fibre cycles of holomorphic maps

This and the second part of the paper are to a great extent parts of my thesis [Si] which also appeared as preprint Mathematica Gottingensis 5/92. This work was partly supported by the S7B170 Geometrie und Analysis in Göttingen     

A note on the classification of holomorphic harmonic morphisms

In this note we give a complete classification of those holomorphic maps :U ⁿ defined on open and connected subsets of ^m which are harmonic morphisms.The first author was supported by the Icelandic Science Fund.     

Polynomial diffeomorphisms ofC 2

Partially supported by NSF grant #DMS-8602020     

Iteration of holomorphic maps on Lie balls

We investigate iterations of fixed-point free holomorphic self-maps on a Lie ball of any dimension, where a Lie ball is a bounded symmetric domain and the open unit ball of a spin factor which can be infinite dimensional. We describe the invariant domains of a holomorphic self-map f on a Lie ball D when f   is fixed-point free and compact, and show that each limit function of the iterates (fⁿ)$(f^n)$ has values in a one-dimensional disc on the boundary of D  . We show that the Möbius transformation _ga$g_a$ induced by a nonzero element a in D may fail the Denjoy–Wolff-type theorem, even in finite dimension. We determine those which satisfy the theorem.     

Condition R and proper holomorphic maps between equidimensional product domains

We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as a product of proper mappings between the factor domains.     

Lifted infinitesimal holomorphic representation for the n-dimensional complex hyperbolic ball and for Cartan domains of type I

The Laplacian and Ornstein–Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U(n,1)$U(n,1)$. We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to extend the results to domains with matrix elements.     

C k -estimates for the -equation on convex domains of finite type

For a bounded convex domain with C^∞ smooth boundary of finite type m and q=1, . . . ,n−1, we construct a -solving integral operator T*_q such that for all k ∈ ℕ and the usual C^k and -norms the operator is continuous.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

On some estimates and Carleson type measure for multifunctional holomorphic spaces in the unit ball

The aim of this paper is to obtain some new estimates for multifunctional holomorphic expressions by using properties of Bergman metric ball. Also we obtain some characterizations of Carleson type measure for some multifunctional holomorphic spaces defined with Bergman metric ball.     

Decompositions of factor maps involving bi-closing maps

We describe all possible decompositions of a finite-to-one factor map : _A S, from an irreducible shift of finite type onto a sofic shift, into two maps =, such that the range of is a shift of finite type, and is bi-closing. We also give necessary and sufficient conditions for to be almost topologically conjugate overS to a bi-closing map.     

Leibniz cohomology and the calculus of variations

A geometric interpretation of the Leibniz coboundary is given in terms of the calculus of variations. For a differentiable manifold M, Leibniz cohomology generalizes de Rham cohomology by including all tensors as cochains. When applied to two-tensors, the conditions for the vanishing of a Leibniz cochain are related to the necessary conditions to achieve an extreme value of the integral of the tensor over an immersed surface. A local formula for the coboundary of any tensor is given in terms of a coordinate chart, and the Leibniz coboundary of the Riemann curvature tensor is computed in terms of the derivative of sectional curvature.     

Homoclinic and heteroclinic orbits for a class of Hamiltonian systems

The existence of a rich structure of homoclinic and heteroclinic solutions is established for a family of Hamiltonian systems that serve as a simpler model for the multiple pendulum system. The proof is based on recently developed arguments from the calculus of variations that have proved useful in finding actual solutions of an equation near approximate solution.This research was sponsorted in part by the National Science Foundation under grant #MCS-8110556 and the U. S. Army Research Office under Contract #DAAL03-87-K-0043. Any reproduction for the purpose of the United States Government is permitted.     

On functions with one‐dimensional holomorphic extension property in circular domains

In this paper we consider continuous functions given on the boundary of a circular bounded domain D in , , and having the one‐dimensional holomorphic extension property along family of complex lines, passing through a finite number of points of D. We study the problem of existence of holomorphic extension of such functions into D.     

Componentwise perturbation bounds for some matrix decompositions

Componentwise perturbation bounds for the Cholesky,LDL ^H ,QR andLU decompositions are derived. The bounds improve known results of the same type and reveal the structural characteristics of the perturbations.This subject was supported by the Institute of Information Processing of the University of Umeå and the Swedish Natural Science Research Council.     

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

Summary A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouché to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.     

Categories of abstract smooth models and their singular envelopes

We define the categories of (abstract) smooth models (Definition 1.2) and, in the additive case, their singular envelopes (Definition 1.5). The first main result is a relative version of the Yoneda representation theorem (Theorem 1.6), and the second one is an existence and uniqueness theorem for the singular envelope (Theorem 1.7). In fact we prove the existence of a canonical process which associates with each additive smooth-model categoryS a singular envelopeS-an ofS, whose objects are calledS-analytic spaces (Definition 5.1). We notice that most of the fundamental categories of geometry are of the formS-an (up to equivalence). As an application we introduce here two such categories: Banach differentiable spaces and Banach mixed spaces.The author is indebted to the Department of Mathematics of the University of Rome (La Sapienza) for hospitality and financial support and to the referees for their critical comments on the first version of this paper.