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Unique determination of local C R -maps by their jets: a survey

Dmitri Zaitsev — 2002

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We survey results on unique determination of local C R -automorphisms of smooth C R -manifolds and of local biholomorphisms of real-analytic C R -submanifolds of complex spaces by their jets of finite order at a given point. Examples generalizing [28] are given showing that the required jet order may be arbitrarily high.

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Wilhelm KaupDmitri Zaitsev — 2006

Journal of the European Mathematical Society

We present a large class of homogeneous 2-nondegenerate CR-manifolds M , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains U , V in M extends to a global real-analytic CR-automorphism of M . We show that this class contains G -orbits in Hermitian symmetric spaces Z of compact type, where G is a real form of the complex Lie group Aut ( Z ) 0 and G has an open orbit that is a bounded symmetric domain of tube type.

Dynamics of one-resonant biholomorphisms

Filippo BracciDmitri Zaitsev — 2013

Journal of the European Mathematical Society

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in C n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

Decomposition of CR-manifolds and splitting of CR-maps

Atsushi HayashimotoSung-Yeon KimDmitri Zaitsev — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing...

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