Sierpinski 地毯上S4模型的临界特性

通过键移动重整化群的方法, 分析了Sierpinski 地毯上S⁴模型的临界行为, 得到了系统的临界点. 由得到的结果可知, 本系统不仅有一个高斯不动点, 而且还存在着一个Wilson Fisher不动点, 把它与Sierpinski 地毯上的高斯模型相互对比, 发现本系统的临界点变化很大. 这说明这两个系统隶属于两个不同的普适类.     

Bounds of the Hausdorff Measure of Sierpinski carpet

By means of the idea of [2]（Jia Baoguo,J.Math.Anal.Appl.In press） and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

Hausdorff dimension and measure of a class of subsets of the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.     

Isoperimetric Estimates on Sierpinski Gasket Type Fractals

For a compact Hausdorff space that is pathwise connected, we can define the connectivity dimension to be the infimum of all such that all points in can be connected by a path of Hausdorff dimension at most . We show how to compute the connectivity dimension for a class of self-similar sets in that we call point connected, meaning roughly that is generated by an iterated function system acting on a polytope such that the images of intersect at single vertices. This class includes the polygaskets, which are obtained from a regular -gon in the plane by contracting equally to all vertices, provided is not divisible by 4. (The Sierpinski gasket corresponds to .) We also provide a separate computation for the octogasket (), which is not point connected. We also show, in these examples, that , where the infimum is taken over all paths connecting and , and denotes Hausdorff measure, is equivalent to the original metric on . Given a compact subset of the plane of Hausdorff dimension and connectivity dimension , we can define the isoperimetric profile function to be the supremum of , where is a region in the plane bounded by a Jordan curve (or union of Jordan curves) entirely contained in , with . The analog of the standard isperimetric estimate is . We are particularly interested in finding the best constant and identifying the extremal domains where we have equality. We solve this problem for polygaskets with . In addition, for we find an entirely different estimate for as , since the boundary of has infinite measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.

     

经典分形集测度上估的计算机搜索Ⅱ──对典型例子Sierpinski垫片计数技术和格点跟踪技术的剖析

A upper estimate function ν(x) of Hausdorff measure Hs (S) ofSierpinski Gasket is given. A mathematical representation of the upperapproximate value νN (x) to ν(x) and a simple algorithm ofνN (x) based on lattice tracing technique are also derived. As asimple corollary, the estimation Hs(S)≤ min ν15 (n·10-5)=ν15 (0.50783)=0.81794 …is obtained.     

A generalized multifractal spectrum of the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which the limiting frequency of a horizontal fibre falls into a prescribed closed interval. We obtain the explicit expression for the Hausdorff dimension of these subsets in terms of the parameters of the construction and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.     

On singularity of energy measures on self-similar sets

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60)     

铌酸锂晶体中的分形几何观察

研究了铌酸锂晶体沿c轴的真实生长界面及腐蚀坑模式,在两种情况中均观察到了明显的塞尔宾斯基(Sierpinski)三角垫分形几何特征,两种情况的分形维度通过计算得出均为ln3/ln2≈1.58.