Martin boundary and exit space on the Sierpinski gasket

We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.     

Energy and Laplacian of fractal interpolation functions

In this paper, we first characterize the finiteness of fractal interpolation functions(FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket(SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5: Δu = 0 on SG\{q_1, q_2, q_3}, and u(q_i) = a_i, i = 1, 2, 3, where q_i, i = 1, 2, 3, are boundary points of SG.     

SOME PATH PROPERTIES OF BROWNIAN MOTION AND SUPER-BROWNIAN MOTION ON THE SIERPINSKI GASKET

1SomeNotationsLetB(t)betheBrownianmotionontheSierpinskigasket.ThisprocesswasfirstconstructedbyM.T.BallotandE.A.Perkins[2],sowewillsdoptthenotationstherein.Forpurposeofsyste~izationsomeboortantnotationswillbereviewedbriefly.LetaO=(0,0),al=(1,0)andaZ==(i,4),letFO={ac,al,a2}bethevenicesOfanequilateraltriangleintheplaneofsideone.DefineinductivelyLetG6=U7=.Fi'GObeGStogetherwithitsreflectioninthey-axis.LetthenG=of(Goo),theclosureOfG.,isnamedtheSierpinskigasket.NowweintroduceG(u):LetG(o…     

Maximally singular Sobolev functions

It is known that for any Sobolev function in the space Wm,p(RN), p?1, mp?N, where m is a nonnegative integer, the set of its singular points has Hausdorff dimension at most N−mp. We show that for p>1 this bound can be achieved. This is done by constructing a maximally singular Sobolev function in Wm,p(RN), that is, such that Hausdorff's dimension of its singular set is equal to N−mp. An analogous result holds also for Bessel potential spaces Lα,p(RN), provided αp<N, α>0, and p>1. The existence of maximally singular Sobolev functions has been announced in [Chaos Solitons Fractals 21 (2004), p. 1287].     

一类广义Sierpinski海绵的填充测度

对于m2的任何整数m,本文定义了R~m中的一类广义 Sierpinski海绵,同时给出了该类海绵的填充测度的计算公式.     

Sierpinski垫片Hausdorff测度的上方估值

Sierpinski垫片S是三个经典的自相似集之一具有Housdorff维数     

Bounds of Hausdorff measure of the Sierpinski gasket

By a new method, we obtain the lower and upper bounds of the Hausdorff measure of the Sierpinski gasket, which can approach the Hausdorff measure of the Sierpinski gasket infinitely.     

一类Sierpinski地毯的分细分析方法和其Hausdorff测度准确值

计算一类Sierpinski地毯的Hausdorff测度,利用投影降低分形维数及分形的分细分析方法,得到了这一类Sierpinski地毯的Hausdorff测度准确值.当这类Sierpinski地毯的Hausdorff维数s∈[log54,1]时,有Hausdorff测度准确值为Hs(S)=1.     

关于两篇有关自相似分形集的Hausdorff测度的论文的注记

指出了发表在《数学的实践与认识》上的两篇关于Sierpinski垫片和Sierpinski锥的Hausdorff测度的论文中主要结论的错误所在,并从若干方面讨论了相关的几个问题.