The complement of certain recursively defined sets

An RD-set (recursively defined) is a minimal set of positive integers containing a given seed and closed under a given set of unary linear operations (x → ax + b). We examine conditions under which the complement of an RD-set may be expressed as the disjoint union of infinite arithmetic progressions.     

The generalized corona theorem

The results of this paper have been presented by the second author at the joint AMS-DMV Meeting 1993 in Heidelberg (Germany)     

Stable intersections of affine cantor sets

We construct open sets of pairs of affine Cantor sets (K, K′) with the simplest possible combinatorics (i. e., defined by two affine increasing maps) which have stable intersection, while the product of lateral thicknesses τ_R(K) • τ_L (K′) is smaller than one. Thus, in a strong form, the converse to the following classic result due to Newhouse is not true: if the product of the thicknesses of two Cantor sets is bigger than one then there are translations of them which have stable intersection. We also describe the topological structure of K—K′ for typical pairs (K, K′) in this open set.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

两个齐次Cantor集的算术和

§ 1　IntroductionThe book[1 ] and the references therein show thatthe structure of arithmetic sums ofCantor sets is relevantto natural questions in smooth dynamics.Palis and Takens[1 ] askedabout the structure of the sums of two Cantor sets and conjectured that“typically” theyhave either zero Lebesgue measure or contained intervals. In 1 997,Solomyka[2 ] showedthatfor eachγ∈ 0 ,12 ,the set Kγ+Kλ(where Kλ,Kγis the middle-α Cantorset forα=1 -2λ or 1 -2γ) of two centered Cantor s…     

On the magnification of cantor sets and their limit models

For aC ¹⁺ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC ¹⁺ diffeomorphic withC (for some >0, (0,1)), but not all suchC ¹⁺ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's scaling function. An alternative definition of a fractal is discussed.With 1 Figure     

On dimensions and measures of statistically constructed cantor sets

We investigate the Hausdorff dimension and the packing dimension of random Cantor sets. That is, using the Gibbs measures, we can conclude that in our Cantor sets the Hausdorff dimension coincides with the packing dimension and this common value is characterized as the unique zero point of a certain function. A striking difference from deterministic cases appears when we consider measures of these sets. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Hausdorff measures for a class of homogeneous cantor sets

We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.     

A characterization theorem for certain unions of two starshaped sets in R 2

Let S be a compact set in R ². For S simply connected, S is a union of two starshaped sets if and only if for every F finite, F bdry S, there exist a set G bdry S arbitrarily close to F and points s, t depending on G such that each point of G is clearly visible via S from one of s, t. In the case where S has at most finitely many components, the necessity of the condition still holds while the sufficiency fails.     

Generalized Douglas algebras and the corona theorem

Donetsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 37–42, January–February, 1991.     

Oscillatory Riemann-Hilbert problems and the corona theorem

The paper is devoted to the Riemann-Hilbert problem with matrix coefficient having in Hardy spaces [H_p^±]²,1<p?∞, on half-planes . Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann-Hilbert problem in [H_∞^±]² we study the solvability of the non-homogeneous Riemann-Hilbert problem in [H_p^±]²,1<p<∞, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in and ). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H_∞⁺ are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.     

The projection theorem for spectral sets

LetG be a locally compact group with polynomial growth and symmetricL ¹-algebra andN a closed normal subgroup ofG. LetF be a closedG-invariant subset of Prim_* L ¹(N) andE={ker; with |N(k(F))=0}. We prove thatE is a spectral subset of Prim_* L ¹(G) ifF is spectral. Moreover we give the following application to the ideal theory ofL ¹(G). Suppose that, in addition,N is CCR andG/N is compact. Then all primary ideals inL ¹(G) are maximal, provided allG-orbits in Prim_* L ¹(N) are spectral.Dedicated to Professor Elmar Thoma on the occasion of his 60th birthday     

Multipliers of Qs spaces and the corona theorem

For s>0, let Q_s be the space of all analytic functions onthe unit disc such that |f'(z)|²(1–|z|²)^s dA(z) isan s-Carleson measure. Here we prove that the corona theoremholds for the algebra of pointwise multipliers of Q_s.