Level sets of harmonic functions on the Sierpiński gasket

Abstract

We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpiński gasket. We introduce a parameter, called eccentricity, which classifies these functions up to affine transformations h→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.