Bounded holomorphic functions and maps with some boundary behavior

Let Ω be a domain of . As a study of boundary behavior of functions and maps in Ω, we consider “the linear cluster set,” which is the cluster set along segment terminating at a boundary point of Ω. We prove that there exist bounded holomorphic functions and maps defined in Ω which have the linear cluster sets of positive measure at every point of a discrete subset of the boundary of Ω under some conditions.     

An example in holomorphic fixed point theory

If is the open unit ball in the Cartesian product furnished with the -norm , where and , then a holomorphic self-mapping of has a fixed point if and only if for some

     

Zeroes of complete polynomial vector fields

We prove that a complete polynomial vector field on has at most one zero, and analyze the possible cases of those with exactly one which is not of Poincaré-Dulac type. We also obtain the possible nonzero first jet singularities of the foliation at infinity and the nongenericity of completeness. Connections with the Jacobian Conjecture are established.

     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

The Dimension Function of Holomorphic Spaces of a Real Submanifold of an Almost Complex Manifold

Let M be a real submanifold of an almost complex manifold and let be the maximal holomorphic subspace, for each x M. We prove that c:M,c(x)=dim H _x is upper-semicontinuous.     

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the holomorphic K-theory of a projective variety. This K-theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory is built out of studying algebraic bundles over a variety up to algebraic equivalence. In this paper we will give calculations of this theory for flag like varieties which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors we will show that there is a rational isomorphism of graded rings between holomorphic K-theory and the appropriate morphic cohomology groups, in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the symmetrized loop group U(n)/ _n where the symmetric group _n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then rationally this is isomorphic to topological K-theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag mani-folds.     

Unbounded convex mappings of the ball in

In this paper, we study univalent holomorphic mappings of the unit ball in that have the property that the image contains a line for some , . We show that under certain rather reasonable conditions, up to composition with a holomorphic automorphism of the ball, the mapping is an extension of the strip mapping in the plane to higher dimensions.

     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

On the ε-entropy of classes of holomorphic functions

Suppose thatB _R ^d is a ball of radiusR in ℂ ^d and σ is the standard measure on the unit sphere in ℂ ^d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH _R ⁰ (l, p, d) we denote the class of functionsf holomorphic inB _R ^d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH ^p (B _R ^d ). In this paper an asymptotic formula for the ε-entropy of the classH _R ⁰ (l, p, d) in the spacesL ^p (σ), 1≤p<∞, and is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 286–293, August, 2000.     

A Note on the KAM Theorem for Symplectic Mappings

The mapping version of Pöschel's theory on differentiable foliation structures of invariant tori is presented and the relevant estimates explicitly in terms of the diophantine constant and the nondegeneracy parameters of frequency maps are given. As a direct application of the main result, a generalization of Moser's small twist theorem to high dimensions is given.