The Hausdorff Dimension of Exceptional Sets Associated with Normal Forms

The Hausdorff dimension is obtained for exceptional sets associatedwith linearising a complex analytic diffeomorphism near a fixedpoint, and for related exceptional sets associated with obtaininga normal form of an analytic vector field near a singular point.The exceptional sets consist of eigenvalues which do not satisfya certain Diophantine condition and are ‘close’to resonance. They are related to ‘lim-sup’ setsof a general type arising in the theory of metric Diophantineapproximation and for which a lower bound for the Hausdorffdimension has been obtained.     

Measures and their dimension spectrums for cookie-cutter sets in ℝd

The characteristics of cookie-cutter sets in ℝ^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly α-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved. This project is supported by the National Natural Science Foundation of China.     

Hausdorff dimension of attractors for two dimensional Lorenz transformations

A class of transformations on [0, 1]², which includes transformations obtained by a Poincare section of the Lorenz equation, is considered. We prove that the Hausdorff dimension of the attractor of these transformations equalsz+1 wherez is the unique zero of a certain pressure function. Furthermore we prove that all vertical intersections with this attractor, except of countable many, have Hausdorff dimensionz.     

Anisotropic branching random walks on homogeneous trees

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff dimension of Λ∩Ω_μ, where Ω_μ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ω_μ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ω_μ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point. Received: 30 June 1998 / Revised version: 10 March 1999     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L^∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set E_f of B-points is everywhere dense and has the maximal Hausdorff dimension, dim _H(E_f ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 219–236.     

Hausdorff measures of a class of Sierpinski carpets

thenandIn this paper, a lemma as a new method to calculate the Hausdorff measure of fractal is given. And the exact values of Hausdorff measure of a class of Sierpinski sets which satisfy balance distribution ang dimension ≤1 are obtained     

齐次完全集的拟对称极小性

作为Cantor型集的推广,文志英和吴军引入了齐次完全集的概念,并基于齐次完全集的基本区间的长度以及基本区间之间的间隔的长度,得到了齐次完全集的Hausdorff维数.本文研究齐次完全集的拟对称极小性,证明在某些条件下Hausdorff维数为1的齐次完全集是1维拟对称极小的.     

An extension of the Weil–Petersson metric to quasi-Fuchsian space

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmüller space. We use this to describe a metric on Teichmüller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson metric of the associated earthquake vector of the lamination. Martin Bridgeman research supported in part by NSF grant DMS 0305634. Edward C. Taylor research supported in part by NSF grant DMS 0305704.     

Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.     

平面区域拟共形变换可微性的可去集

本文研究平面区域上K-qc映射的不可微集合的Hausdorff维数.对任何K>1,给出了平面区域上一个具体的K-qc映射,它的不可微集合的Hausdorff维数为2.     

The Integral Formula for Calculating the Hausdorff Measure of Some Fractal Sets

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of Mass Distribution, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.     

广义α-Stable过程的像集和图集的一致维数

研究了未必具有随机一致Holder条件的N指标d维广义α-stable过程的像集和图集的一致维数问题,并在一定条件下得到了N指标d维广义α-stable过程像集约一致Hausdorff维数和一致Packing维数的上、下界,图集的一致Hausdorff维数和一致Packing维数的上界,包含了多指标α-stable过程和广义布朗单相应的结果．     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Weak tangent and level sets of Takagi functions

In this paper, we study some properties of Takagi functions and their level sets. We show that for Takagi functions $$T_{a,b}$$ with parameters a, b such that ab is a root of a Littlewood polynomial, there exist large level sets. As a consequence, we show that for some parameters a, b, the Assouad dimension of graphs of $$T_{a,b}$$ is strictly larger than their upper box dimension. In particular, we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.     

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001     

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.     

Relations among whitney sets,self-similar arcs and quasi-arcs

We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent. Supported by Special Funds for Major State Basic Research Projects of China, Morningside Center of Mathematics, NSFC (No. 10241003) and ZJNFS (No. 101026).     

Distributional Dimension of Fraetal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension     

Self-similar sets in complete metric spaces

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.