Isoperimetric Estimates on Sierpinski Gasket Type Fractals
Isoperimetric Estimates on Sierpinski Gasket Type Fractals
复制标题
DOI：
10.1090/s0002-9947-99-01999-6
复制
发表时间：
1999-01
影响因子：
1.3
通讯作者：
R. Strichartz
中科院分区：
文献类型：
--
作者：
R. Strichartz
对于路径连通的紧豪斯多夫空间 F，我们可以将连通维数 β 定义为所有 b 的下确界，使得 F 中的所有点都可以通过一条最多为 b 的豪斯多夫维数的路径连接。我们展示了如何计算 Rn 中一类自相似集的连通性维数，我们称之为点连接，大致意思是 F 是由作用于多面体 P 的迭代函数系统生成的，使得 P 的图像在单个顶点相交。此类包括聚垫片，这些垫片是通过均匀收缩到所有 n 个顶点而从平面中的正 n 边形获得的，前提是 n 不能被 4 整除。（Sierpinski 垫片对应于 n = 3。）我们还为八角垫片 (n = 8) 提供单独的计算，它不是点连接的。在这些例子中，我们还表明， infHβ(γx,y) ，其中下确界是在连接 x 和 y 的所有路径 γx,y 上获取的，并且 Hβ 表示豪斯多夫测度，相当于 F 上的原始度量。给定 Hausdorff 维数 α 和连通维数 β 的平面的紧凑子集 F，我们可以将等周剖面函数 h(L) 定义为 Hα(F ∩D) 的上界，其中 D 是平面中由 Jordan 曲线（或 Jordan 曲线并集）γ 完全包含在 F 中的区域，其中 Hβ(γ) ≤ L。标准等周估计的模拟是 h(L) ≤ cLα/β 。我们特别感兴趣的是寻找最佳常数 c 并确定我们具有相等性的极值域。我们针对 n = 3, 5, 6, 8 的聚垫片解决了这个问题。此外，对于 n = 5, 6, 8，我们发现 h(L) 的估计完全不同，因为 L→∞，因为 F 的边界具有无限的 Hβ 测量值。我们发现等周轮廓函数是不连续的，并且极值域具有相对简单的多边形边界。我们简要讨论谢尔宾斯基垫片的最小路径的性质，以及内在度量中的等径问题。
For a compact Hausdorff space F that is pathwise connected, we can define the connectivity dimension β to be the infimum of all b such that all points in F can be connected by a path of Hausdorff dimension at most b. We show how to compute the connectivity dimension for a class of self–similar sets in Rn that we call point connected , meaning roughly that F is generated by an iterated function system acting on a polytope P such that the images of P intersect at single vertices. This class includes the polygaskets, which are obtained from a regular n–gon in the plane by contracting equally to all n vertices, provided n is not divisible by 4. (The Sierpinski gasket corresponds to n = 3.) We also provide a separate computation for the octogasket (n = 8), which is not point connected. We also show, in these examples, that infHβ(γx,y) , where the infimum is taken over all paths γx,y connecting x and y, and Hβ denotes Hausdorff measure, is equivalent to the original metric on F . Given a compact subset F of the plane of Hausdorff dimension α and connectivity dimension β, we can define the isoperimetric profile function h(L) to be the supremum of Hα(F ∩D), where D is a region in the plane bounded by a Jordan curve (or union of Jordan curves) γ entirely contained in F , with Hβ(γ) ≤ L. The analog of the standard isperimetric estimate is h(L) ≤ cLα/β . We are particularly interested in finding the best constant c and identifying the extremal domains where we have equality. We solve this problem for polygaskets with n = 3, 5, 6, 8. In addition, for n = 5, 6, 8 we find an entirely different estimate for h(L) as L→∞, since the boundary of F has infinite Hβ 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.