Constructing a Laplacian on the Diamond Fractal

Abstract

Kigami has shown how to construct Laplacians on certain self-similar fractals, first for the Sierpiński gasket and then for the class of postcritically finite (\pcf\) fractals, subject to the solution of certain algebraic equations. It is desirable to extend this method to as large a class of fractals as possible, so in this paper we examine a specific example that exhibits features associated with finite ramification, but which does not fall into the class of \pcf\ fractals. We show by a method of deconstruction that this fractal is a member of a family of three fractals for which the \pcf\ condition holds in a generalized sense. We then study the algebraic equations whose solution is required for the actual construction of the Laplacian. We obtain experimental evidence for the existence and uniqueness of solutions. This experimental work uncovers two symmetries that were not initially apparent, only one of which has a natural explanation. By exploiting the symmetries, we give a nonconstructive proof of existence.