Cantor集的自乘积集的Hausdorff测度的下界

证明了三分Cantor集C的自乘积集C×C的Hausdorff测度满足Hlog34(C×C)≥1.48329.     

Cantor集的性质及应用

Cantor集是实函数论中一类重要的集合.本文从定义、性质及应用三方面研究了Cantor集.目的是帮助初学者对Cantor集有一个较全面的认识.     

三分Cantor集自乘积的Hausdorff测度的估计

本文证明了三分Cantor集C自乘积集C×C的Hausdorff测度,满足1≤H~((log_3)~4)(C×C)≤1.502879.     

Cantor集的自乘积集的Hausdorff测度的下界

证明了三分Cantor集C的自乘积集C×C的Hausdorff测度满足 H~(log_3)~4(C×C)≥1.48329。     

网测度及应用

给出了n-进制网测度一个具体的定义,并证明了它的一些性质如:外测度、度量外测度.给出网测度的质量分布原理,证明了它与Hausdorff测度等价,并计算了直线上的三分Cantor集和Sierpinski地毯的网测度.     

关于Cantor三分集定义方式的一个新进展

通过对Cantor三分集本质结构进行深刻洞察,并结合结构化方法对Cantor三分集的定义方式进行研究.定义了?上的无再生三分族、Cantor三分族等概念,证明了Cantor三分集的一个非三进制数模式的新公式,并证明了一个关于三进制数的新公式.为Cantor三分集的"三进制数定义法"提供了一种严密的理论基础,同时也增进了...     

一类直线上自相似集的相似压缩不动点的分布

先构造了一类直线上的自相似集,研究了它的相似压缩不动点的坐标公式.作为推论我们给出了三分Cantor集相似压缩不动点的坐标公式,从而首次发现了它的相似压缩不动点的分布规律.     

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。     

齐次Cantor集的填充测度(英文)

本文给出了一个计算齐次Cantor集的填充测度的公式.     

m分非均匀Cantor集的Hausdorff测度

证明了m分非均匀Cantor集的E的H ausdorff测度HS(E)=1.     

Hausdorff dimension of perturbed Cantor sets without some boundedness condition

A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Measure-Preserving Shift on the Cantor Set

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Cantor尘的Hausdorff测度估计

Cantor dust is a classical fractal set .It is very important to compute its Hausdorff measure.In this Paper,we obtained the expression and approximation of Hausdorff measure of Cantor dust through the analysis of interpolation function of a certain of mass distribution defined on Cantor dust.     

Semigroups whose Julia sets are Cantor targets

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2.     

含参变量Cantor集的Hausdorff测度

本文研究了菱形为基本集所构成的的广义Cantor集的Hausdorff测度问题.利用菱形几何结构的相关证明方法,获得了此类广义Cantor集的Hausdorff测度准确值,推广了曾超益和许绍元等人的已有结果.     

Cantor families of periodic solutions for wave equations via a variational principle

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities.     

Simple geodesics and a series constant over Teichmuller space

We investigate the Birman Series set in a neighborhood of a cusp on a punctured surface, showing that it is homeomorphic to a Cantor set union countably many isolated points cross a line. The local topology of the Cantor set is shown to be related in a simple way to the global behavior of simple geodesics. From this we deduce that a certain series is constant across the Teichmuller space. Oblatum IV-1995 & 5-VI-1997     

Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle

Avila recently introduced a new method for the study of the discrete Schrödinger operator with limit-periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit-periodic Verblunsky coefficients. Specifically, I represent these two-sided Verblunsky coefficients as a continuous sampling of the orbits of a Cantor group by a minimal translation. I then investigate the measures that arise on the unit circle as I vary the sampling function. I show that generically the spectrum is a Cantor set and we have empty point spectrum. Furthermore, there exists a dense set of sampling functions for which the corresponding spectrum is a Cantor set of positive Lebesgue measure, and all corresponding spectral measures are purely absolutely continuous.     

Exact packing measure of central Cantor sets in the line

In this paper we consider a class of symmetric Cantor sets in $\mathbb{R}$. Under certain separation condition we determine the exact packing measure of such a Cantor set through the computation of the lower density of the uniform probability measure supported on the set. With an additional restriction on the dimension we give also the exact centered Hausdorff measure by computing the upper density.