Related papers: Renormalization of $C^r$ H\'enon map : Two dimensi…

For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is…

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear…

An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a…

We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic surfaces containing C^2 for which the curve at infinity is irreducible), compute the moduli space of these surfaces and their groups of auomorphisms. In…

For a two parameter family of two-dimensional piecewise linear maps and for every natural number $ n $ we prove not only the existence of intervals of parameters for which the respective maps are $ n $ times renormalizable but also we show…

As is well-known, there exist nonconstant holomorphic maps from the plane into the Riemann sphere $\PP^1$ minus two points, the simplest example of which is an explicit realization of the uniformization map given by applying the exponential…

In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $\Omega$ and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$…

For a compact set $E \subset \mathbb{C}$ containing more than two points, we study asymptotic behavior of normalized zero counting measures $\{\mu_k \}$ of the derivatives of Faber polynomials associated with $E$. For example if $E$ has…

Let G be a connected reductive group (over $\mathbb{C}$) and H a connected semisimple subgroup. The dimension data of H (realative to its given embedding in G) is the collection of the numbers $\{{\rm dim} V^{H}\}$, where V runs over all…

We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the…

A number of 2d and 3d four-fermion models which are renormalizable ---in the $1/N$ expansion--- in a maximally symmetric constant curvature space, are investigated. To this purpose, a powerful method for the exact study of spinor heat…

This paper presents a new concept of geometric multipole expansion for the conductivity or anti-plane elasticity problem in two dimensions by using the Faber polynomials. As an application, we construct semi-neutral inclusions of general…

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let $\f$ be its extension to a bundle $\E = X \times\Rm$ by smooth fiber maps $\f_x : \E_x \to \E_{fx}$ so that the derivative of $\f$ at the zero section has…

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of…

We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary…

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski…

In this work we study the local structure of analytic planar vector fields that are reversible with respect to the linear involution $R(u,v)=(u,-v)$. We show that every analytic reversible vector field with a nondegenerate equilibrium is…

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point,…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…