Spectral analysis on Barlow and Evans’ projective limit fractals

Abstract

We review the projective limit construction of a state space for a Markov process use by Barlow and Evans. On this state space we construct a projective limit Dirichlet form in a process analogous to Barlow and Evan’s construction of a Markov process. Then we study the spectral properties of the corresponding Laplacian using the projective limit construction. For some examples, such as the Laakso spaces and a Sierpiński pâte à choux, one can develop a complete spectral theory, including the eigenfunction expansions that are analogous to Fourier series. In addition, we construct connected fractal spaces isospectral to the fractal strings of Lapidus and van Frankenhuijsen. Our work is motivated by recent progress in mathematical physics on fractals.