Harmonic Structure on Modified Sierpinski Gaskets

Abstract

How to construct harmonic structures on a class of special self-similar fractals, and then discuss their regularity are important problems in analysis on fractals. J Kigami [3,4,5,6,7] and Strichartz[8,9,10,11] have dicussed in detail. It’s very difficult to built the concept of derivative on fractals directly, therefore we have to consider to construct Laplacians on fractals. The key idea of constructing a Laplacian on fractals is finding a “self-similar” compatible sequence of resistance netwoks on {V m } m ≥ 0. We can start from finite set to built a compatible sequence, then construct a harmonic structure and thus extend to the infinite points. By this way, we could discuss the property of harmonic structures on fractals. In this paper, we study harmonic extension algorithm (matrices) and harmonic structures on modified Sierpinski gaskets (MSG for short). And we also study the relationship between regular harmonic structure and renormalization factor on MSG.KeywordsHarmonic FunctionSierpinski GasketHarmonic StructureHarmonic ExtensionFractal InterpolationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.