Iteration of holomorphic maps in Hilbert spaces

Previous work (Gong-ning Chen, J. Math. Anal. Appl.98 (1984), 305–313) on iteration of holomorphic maps of Cⁿ is continued. The purpose of this note is to extend results given in the above mentioned reference to the case of complex Hilbert spaces. Other comments are appended.     

Critical orbits of holomorphic maps on projective spaces

We study the dynamics of iterated holomorphic maps of a complex projective space onto itself. Relations between the Fatou set and the orbits of critical points are investigated. In particular, results concerning critically finite maps on the Riemann sphere are generalized to higher dimensional case.     

On continuous extensions of grafting maps

The definition of the grafting operation for quasifuchsian groups is extended by Bromberg (preprint) to all -groups. In this paper, we show that the extended grafting maps behave as continuous maps for every sequence which converges ``standardly' to a boundary group, although the maps are not continuous in general. As a consequence of this result, we extend Goldman's grafting theorem for quasifuchsian groups to all boundary -groups.

     

Random iterations of holomorphic maps in complex Banach spaces

Conditions guaranteeing the uniform convergence to constant maps of random iterations of holomorphic contractions on unbounded domains in complex Banach spaces are established.

     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G_δ-measurability (the preimage of each open set is of type G_δ). Also under Martin Axiom, we construct a G_δ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.     

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

Local imbedding of Einstein spaces

Summary A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces. Entrata in Redazione il 23 giugno 1971.     

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space (X, ν) determines a not necessarily topological closure operator c_ν on X. The space (X, ν) is said to be complete if every cluster on (X, ν) is contained in an adherence grill on (X, c_ν). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T₁ closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T₁ Riesz merotopic spaces is epireflective in the category of separated T₁ Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T₁ Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T₁ Riesz merotopic spaces, the reflection of an object being the strict completion.     

Almost holomorphic extensions of ultradifferentiable functions

For a smooth functionf on ℝ ⁿ , we construct an extensionF to ℂ ⁿ with vanishing to a high order on ℝ ⁿ and give precise estimates of how the degree of smothness is reflected in the degree of vanishing. This analysis is used to define the operator on (n,n−1) forms with singularities on ℝ ⁿ .     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.