IUMJ

Title: Limit chains on the Sierpinski gasket

Authors: Michael Hinz

Issue: Volume 60 (2011), Issue 5, 1797-1830

Abstract:

The analysis of the Hodge-Laplacian for differential $p$-forms on smooth Riemannian Manifolds is a well-known theory. The present paper proposes some fractal analog for the case $p = 1$ on the Sierpinski gasket. More precisely, we introduce a new energy functional on substitutes of $1$-forms. It is constructed along a sequence of two-dimensional cell complexes that approximate the Sierpinski gasket 'from above'. The basic techniques resemble ideas known from the analysis of function Laplacians on p.c.f. fractals. We prove the existence of an infinite-dimensional space of limit $1$-chains having positive and finite energy.