On the Hausdorff Dimension of Some Random Cantor Sets

We consider Cantor sets C _X,p constructed by means of a sequence {X _k ^–p } _k=1 of random variables. We find sufficient conditions for the inequality to hold.     

On the Continuity of Julia Sets and Hausdorff Dimension of N CP and Parabolic Maps

Denote by HD(J(f))the Hansdorff dimension of the Julia set J(f)of a rational function f.Our first result asserts that if f is an NCP map,and f_n→f horocyclically, preserving sub-critical relations,then f_n is an NCP map for all n(?)0 and J(f_n)→J(f)in the Hausdorff topology.We also prove that if f is a parabolic map and f_n is an NCP map for all n(?)0 such that f_n→f horocyclically,then J(f_n)→J(f)in the Hansdorff topology, and HD(J(f_n))→HD(J(f)).     

RN中某些分形子集的Hausdorff维数与Hausdorff测度

给出了RN中某些分形子集的Hausdorff维数及其Hausdorff测度估计式.     

满Parry测度的有限型子集的Hausdorff维数与测度

All the full Parry measure subsets of a given subshift of finite type determined by an irreducible 0-1 matrix have the same Hausdorrf dimension and Hausdorff measure which coincide with those of the set of finite type.     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

Hausdorff Dimension of the Set of Generic Points for Gibbs Measures

For a Gibbs measure on the configuration space of a finite spin lattice system, we find (in terms of entropy) the Hausdorff dimension of the set of generic points. Using this result, we evaluate the Hausdorff dimension of level sets for Birkhoff ergodic averages of some continuous functions on the configuration space.     

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

We show that a discrete, quasiconformal group preserving ⁿ has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to ⁿ . This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on ⁿ , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.     

关于自相似集的Hausdorff测度的一个判据及其应用

讨论了满足开集条件的自相似集。对于此类分形，用自然覆盖类估计它的Hausdorff测度只能得到一个上限，因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用，统一处理了一类自相似集，得到了平面上的一个Cantor集－Cantor尘的Hausdorff测度的准确值，并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。     

On the Connection between the Order of Riemann-Liouvile Fractional Calculus and Hausdorff Dimension of a Fractal Function

This paper investigates the fractal dimension of the fractional integrals of a fractal function.It has been proved that there exists some linear connection between the order of Riemann-Liouvile fractional integrals and the Hausdorff dimension of a fractal function.     

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.     

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.     

On Fraïssé’s conjecture for linear orders of finite Hausdorff rank

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ₂(0), the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in  +“φ₂(0) is well-ordered” and, over , implies  +“φ₂(0) is well-ordered”.     

Estimates for the Norm of Green's Matrix Based on the Integral Performance Criterion for Dichotomy and Hausdorff Set Bounds

For a system of ordinary differential equations of the form , , we obtain new upper bounds for the second norm of Green's matrix using the integral performance criterion for dichotomy and bounds of the Hausdorff set of the matrix A. These estimates are considerably better for many applications than some well-known ones.     

Upper and Lower Semicontinuity of Solution Sets for Parametric Generalized Vector Quasi-equilibrium Problems

In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.     

Dense Subsets of H1/2(S2, S1)

We prove that the maps from S ² intoS ¹ having a finite number of isolated singularities ofdegree ±1 are dense for the strong topology inH ^1/2(S ², S ¹). We also prove that smooth maps are densein H ^1/2(S ², S ¹)for the sequentially weak topology andthat this is no more the case in H ^s (S ², S ¹) for s> 1/2.     

Berry–Esseen theorem and local limit theorem for non uniformly expanding maps

In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed , and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.