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Rasul Shafikov

Publications and source records attributed to Rasul Shafikov.

45 records · Page 3Linked to original sources
arXiv2006

Real Analytic Sets in Complex Spaces and CR maps

If $R$ is a real analytic set in $\C^n$ (viewed as $\R^{2n}$), then for any point $p\in R$ there is a uniquely defined germ $X_p$ of the smallest complex analytic variety which contains $R_p$, the germ of $R$ at $p$. It is shown that if $R$ is irreducible of constant dimension, then the function $p\to\dim X_p$ is constant on a dense open subset of $R$. As an application it is proved that a continuous map from a real analytic CR manifold $M$ into $\C^N$ which is CR on some open subset of $M$ and whose graph is a real analytic set in $M\times \C^N$ is necessarily CR everywhere on $M$.

math.CV↗
arXiv2006

Boundary regularity of correspondences in $\pmb{{\mbf C}^n}$

Let $M, M'$ be smooth, real analytic hypersurfaces of finite type in ${\mbf C^n}$ and $\h f$ a holomorphic correspondence (not necessarily proper) that is defined on one side of $M$, extends continuously up to $M$ and maps $M$ to $M'$. It is shown that $\h f$ must extend across $M$ as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich--Pinchuk extension result for CR maps.

math.CV↗
arXiv2004

Uniformization of strictly pseudoconvex domains

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.

math.CV↗
arXiv2003

Stable sets, hyperbolicity and dimension

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $Λ$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(Λ)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results in the case of expanding maps.

math.DS↗
arXiv2002

A local extension theorem for proper holomorphic mappings in C^2

Let f be a proper holomorphic mapping between bounded domains D and D' in C^2. Let M, M' be open pieces on the boundaries of D and D' respectively, that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f is contained in M'. It is shown that f extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in C^2.

math.CV↗
arXiv2002

Filtrations, hyperbolicity and dimension for polynomial automorphisms of C^n

In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we obtain a complete description of its orbits. In the second part of the paper we study the Hausdorff and box dimension of the Julia sets of f. We show that the Julia set J has positive box dimension, and (provided f is not volume preserving) that the filled-in Julia set K has box dimension strictly less than 2n. Moreover, if f is hyperbolic, then the Hausdorff dimension of the forward/backward Julia set J^{+/-} is strictly less than 2n.

math.DS↗
arXiv2002

Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in ${\mathbb C}^n$

The following result is proved: Let $D$ and $D'$ be bounded domains in $\mathbb C^n$, $\partial D$ is smooth, real-analytic, simply connected, and $\partial D'$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence $f:D\to D'$ if and only if $\partial D$ and $\partial D'$ a locally CR-equivalent. This leads to a characterization of the equivalence relationship between bounded domains in $\mathbb C^n$ modulo proper holomorphic correspondences in terms of CR-equivalence of their boundaries.

math.CV↗