Diffusions and Laplacians on Laakso, Barlow-Evans, and other fractals

Date of Completion

January 2010






The study of self-adjoint operators on fractal spaces has been well developed on specific classes of fractals, such as post-critically finite and finitely ramified. In Part I, we begin by discussing the spectrum of a self-similar Laplacian on a family of post-critically finite fractals, calculating the spectrum for a general member of this family. To complement this we then discuss a source of post-critically finite fractals from self-similar groups that are associated with the Hanoi Towers game and certain modifications of these groups. Part II develops the spectral analysis of a self-adjoint Laplacian on Laakso spaces. The spectrum of this operator is calculated in general with multiplicities and supported by numerical calculations for many specific Laakso spaces. ^