Isoperimetric Estimates on Sierpinski Gasket Type Fractals
HTML articles powered by AMS MathViewer
- by Robert S. Strichartz PDF
- Trans. Amer. Math. Soc. 351 (1999), 1705-1752 Request permission
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.References
- E. Ayer and R. S. Strichartz, Exact Hausdorff measure and intervals of maximum density for Cantor measures, Trans. Amer. Math. Soc. (to appear)
- Christoph Bandt and Thomas Kuschel, Self-similar sets. VIII. Average interior distance in some fractals, Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 307–317. Measure theory (Oberwolfach, 1990). MR 1183058
- C. Bandt and J. Stahnke, Self–similar sets 6. Interior distance on determinatic fractals, preprint, Greifswald 1990.
- Michael Barnsley, Fractals everywhere, Academic Press, Inc., Boston, MA, 1988. MR 977274
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- S. Havlin and D. Ben–Avraham, Duffision in disordered media, Advances in Physics 36 (1987), 695–798.
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), no. 2, 721–755. MR 1076617, DOI 10.1090/S0002-9947-1993-1076617-1
- Jacques Marion, Mesure de Hausdorff d’un fractal à similitude interne, Ann. Sci. Math. Québec 10 (1986), no. 1, 51–84 (French). MR 841120
- Jacques Marion, Mesures de Hausdorff d’ensembles fractals, Ann. Sci. Math. Québec 11 (1987), no. 1, 111–132 (French, with English summary). MR 912166
- R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
- Umberto Mosco, Variational metrics on self-similar fractals, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 715–720 (English, with English and French summaries). MR 1354712
- Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
- R. S. Strichartz, Piecewise linear wavelets on Sierpinski gasket type fractals, J. Fourier Anal. Appl. 3 (1997), 387–416.
Additional Information
- Robert S. Strichartz
- Affiliation: Department of Mathematics, White Hall, Cornell University, Ithaca, New York 14853
- Email: str@math.cornell.edu
- Received by editor(s): May 14, 1996
- Received by editor(s) in revised form: November 25, 1996
- Published electronically: January 26, 1999
- Additional Notes: Research supported in part by the National Science Foundation, Grant DMS-9623250
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1705-1752
- MSC (1991): Primary 28A80; Secondary 51M16
- DOI: https://doi.org/10.1090/S0002-9947-99-01999-6
- MathSciNet review: 1433127