Approximation of fractals by manifolds and other graph-like spaces

Abstract

We define a distance between energy forms on a graph-like metric measure space and on a discrete weighted graph using the concept of quasi-unitary equivalence. We apply this result to metric graphs and graph-like manifolds (e.g. a small neighbourhood of an embedded metric graph) as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary) and express the distance of the associated energy forms in terms of geometric quantities. We showed in arXiv:1704.00064 that the approximating sequence of energy forms on weighted graphs used in the definition of an energy form on a pcf fractal converge in the sense that the distance in the quasi-unitary equivalence tends to 0. By transitivity of quasi-unitary equivalence, we conclude that we can approximate the energy form on a pcf fractal by a sequence of energy forms on metric graphs and graph-like manifolds. In particular, we show that there is a sequence of domains converging to a pcf fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. Quasi-unitary equivalence of energy forms implies a norm estimate for the difference of the resolvents of the associated Laplace operators. As a consequence, suitable functions of the Laplacians are close resp. converge as well in operator norm, e.g. the corresponding heat operators and spectral projections. The same is true for the spectra and the eigenfunctions in all above examples.