Bounded type Siegel disks of finite type maps with few singular values Dedicated to the Memory of Professor Lei Tan

Let U be an open subset of the Riemann sphere C. We give sufficient conditions for which a finite type map f : U →C with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-(S′)wia tek. We also give sufficient conditions for which, instead, ? has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-(S′)wia tek.     

All bounded type Siegel disks of rational maps are quasi-disks

We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan’s conjecture in the case of bounded type rotation numbers.     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

Relatively compact Siegel disks with non-locally connected boundaries

We construct holomorphic maps f with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of \mathbb C{\mathbb C}.     

Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.     

The Siegel norm,the length function and character values of finite groups

In this paper we present some new results on the connection between the Siegel norm, the length function and irreducible character values of finite groups. In addition, we provide algorithms to compute the length of a cyclotomic integer and the set of cyclotomic integers with Siegel norm bounded by a given positive real number.     

Dimension of the boundary of quasiconformal Siegel disks

We study quasiconformal Siegel disks with critical points in their boundaries. The main result asserts that every subarc of the boundary of the Siegel disk has the Hausdorff dimension strictly larger than 1 and that the boundary does not have a tangent at any point. Oblatum 19-V-2000 & 4-X-2001?Published online: 18 January 2002     

A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.     

Upper bound for the size of quadratic Siegel disks

If is an irrational number, we let {p_n/q_n}_n0, be the approximants given by its continued fraction expansion. The Bruno series B() is defined as

Estimates for the singular values of stieltjes type transforms

We obtain two-sided estimates for the Schatten-von Neumann norms of a selfadjoint Stieltjes type transform in the space of square integrable functions.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Difference sets with few character values

The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd \((v,n)>1\) . It is remarkable that all the known difference sets with gcd \((v,n)>1\) have the so-called character divisibility property. Jungnickel and Schmidt (Difference sets: an update. London Math. Soc. Lecture Note Ser., vol. 245, pp. 89–112, Cambridge University Press, Cambridge 1997) posed the problem of constructing difference sets with gcd \((v,n)>1\) that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.     

Operator spaces with few completely bounded maps

We construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable 1-Hilbertian operator space X₀ such that, whenever X is an infinite dimensional quotient of X₀, X is a subspace of X, and is a completely bounded map, then T=I_X+S, where S is compact Hilbert-Schmidt and ||S||₂/16||S||_cb||S||₂. Moreover, every infinite dimensional quotient of a subspace of X₀ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property. Mathematics Subject Classification (2000):The first author was supported in part by the NSF grants DMS-9970369, 0296094, and 0200714.     

Optimal Gerschgorin‐type inclusion intervals of singular values

In this paper, some optimal inclusion intervals of matrix singular values are discussed in the set Ω_A of matrices equimodular with matrix A. These intervals can be obtained by extensions of the Gerschgorin‐type theorem for singular values, based only on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that upper bounds of these intervals are optimal in some cases and lower bounds may be non‐trivial (i.e. positive) when PA is a H‐matrix, where P is a permutation matrix, which improves the conjecture in Reference (Linear Algebra Appl 1984; 56 :105‐119). Copyright © 2006 John Wiley & Sons, Ltd.     

John disks and extension of maps

We show that a quasisymmetric map between the boundaries of two John disks can be extended to a quasiconformal map of the extended plane. Additional results on John disks are also given.With 2 Figures     

Axial maps with finite supports

We work on axial maps of ω ² and characterize those maps of ω ² that can be represented as a composition of a finite number of axial maps with finite support on each axis.     

Regularity of CR maps between convex hypersurfaces of finite type

We study rigidity and regularity properties of CR maps between smooth convex hypersurfaces of finite type in