Isoperimetric Estimates on Sierpinski Gasket Type Fractals
Author:
Robert S. Strichartz
Journal:
Trans. Amer. Math. Soc. 351 (1999), 17051752
MSC (1991):
Primary 28A80; Secondary 51M16
Published electronically:
January 26, 1999
MathSciNet review:
1433127
Fulltext PDF Free Access
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Abstract: 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 selfsimilar 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.
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Additional Information
Robert S. Strichartz
Affiliation:
Department of Mathematics, White Hall, Cornell University, Ithaca, New York 14853
Email:
str@math.cornell.edu
DOI:
http://dx.doi.org/10.1090/S0002994799019996
PII:
S 00029947(99)019996
Keywords:
Isoperimetric estimates,
Sierpinski gasket,
fractals,
connectivity dimension
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 DMS9623250
Article copyright:
© Copyright 1999 American Mathematical Society
