, Volume 40, Issue 2, pp 335-362

Level sets of harmonic functions on the Sierpiński gasket

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Abstract

We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpiński gasket. We introduce a parameter, calledeccentricity, which classifies these functions up to affine transformationsh→ah+b. We describe three (presumably) distinct measures that describe how the eccentricities are distributed in the limit as we subdivide the gasket into smaller copies (cells) and restrict the harmonic function to the small cells. One measure simply counts the number of small cells with eccentricity in a specified range. One counts the contribution to the total energy coming from those cells. And one counts just those cells that intersect a fixed generic level set. The last measure yields a formula for the box dimension of a generic level set. All three measures are defined by invariance equations with respect to the same iterated function system, but with different weights. We also give a construction for a rectifiable curve containing a given level set. We exhibit examples where the curve has infinite winding number with respect to some points.

Part of this research was carried out during a visit to the Mathematics Department of Cornell University.
Research supported in part by the National Science Foundation, grant DMS-9970337.
Research supported by the National Science Foundation through the Research Experiences for Undergraduates (REU) program at Cornell.