Rigidity in holomorphic and quasiregular dynamics

We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.

     

Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented. To cite this article: A. Ibort, D. Martínez Torres, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Rigidity of holomorphic Collet-Eckmann repellers

We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann mapf is topologically conjugate to a holomorphic mapg, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, theng is Collet-Eckmann, too. The first author acknowledges support by Polish KBN Grant 2 P03A 025 12 “Iterations of Holomorphic Functions” and support of the Hebrew University of Jerusalem, where a part of tha paper was written. The second author is grateful for the hospitality and support of the Caltech, where a part of the paper was written.     

Rigidity of certain holomorphic foliations

There is a well-known rigidity theorem of Y. Ilyashenko for (singular) holomorphic foliations in and also the extension given by Gómez-Mont and Ortíz-Bobadilla (1989). Here we present a different generalization of the result of Ilyashenko: some cohomological and (generic) dynamical conditions on a foliation on a fibred complex surface imply the d-rigidity of , i.e. any topologically trivial deformation of is also analytically trivial. We particularize this result to the case of ruled surfaces. In this context, the foliations not verifying the cohomological hypothesis above were completely classified in an earlier work by X. Gómez-Mont (1989). Hence we obtain a (generic) characterization of non-d-rigid foliations in ruled surfaces. We point out that the widest class of them are Riccati foliations.

     

A note on the topological transversality theorem for acyclic maps

A new topological transversality theorem is presented for acyclic maps. The analysis relies on Urysohn’s Lemma and the fact that the unit sphere is contractible in infinite dimensional normed linear spaces.     

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     

Jessen's theorem for holomorphic almost periodic maps

An analog of Jessen's theorem on the existence of a Jessen function and its relation to the distribution of roots of a holomorphic almost periodic function in a strip is obtained for almost periodic maps.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1094–1107, August, 1990.     

A new rigidity result for holomorphic maps

In this paper we determine which vanishing order of a holomorphic map f at a point of the (non necessarily regular) boundary of a very generic domain of is required for f to be constant. In particular this vanishing order is 1 if the boundary is Dini-smooth whereas it is at least β/α if f locally maps a Dini-smooth corner of opening πα into a Dini-smooth corner of opening πβ. Finally an analogous result is stated for the case of a holomorphic map f which maps a cusp into a cusp.     

Projection on Segre varieties and determination of holomorphic mappings between real submanifolds

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Fibre cycles of holomorphic maps

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

Primitive holomorphic maps of curves

Let t: XY be a proper morphism of non-singular irreducible affine curves over . This paper shows that there is seldom a holomorphic function f such that (t,f): X Y × is a holomorphic embedding.     

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