Isoperimetric Estimates on Sierpinski Gasket Type Fractals

Abstract:

For a compact Hausdorff space $F$ that is pathwise connected, we can define the connectivity dimension $\beta$ 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 $\mathbb {R}^{n}$ 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 $\inf \mathcal {H}_{\beta }(\gamma _{x,y})^{1/\beta }$, where the infimum is taken over all paths $\gamma _{x,y}$ connecting $x$ and $y$, and $\mathcal {H}_{\beta }$ denotes Hausdorff measure, is equivalent to the original metric on $F$. Given a compact subset $F$ of the plane of Hausdorff dimension $\alpha$ and connectivity dimension $\beta$, we can define the isoperimetric profile function $h(L)$ to be the supremum of $\mathcal {H}_{\alpha }(F \cap D)$, where $D$ is a region in the plane bounded by a Jordan curve (or union of Jordan curves) $\gamma$ entirely contained in $F$, with $\mathcal {H}_{\beta }(\gamma ) \le L$. The analog of the standard isperimetric estimate is $h(L) \le cL^{\alpha /\beta }$. 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 $L \rightarrow \infty$, since the boundary of $F$ has infinite $\mathcal {H}_{\beta }$ 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.