Combinatorial and analytical problems for fractals and their graph approximations

The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs. The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself. In paper I, we study the number of spanning trees on the sequence of graphs approximating a self-similar set admitting spectral decimation. In paper II, we study harmonic functions on p.c.f. self-similar sets. Unlike the standard Dirichlet problem and harmonic functions in Euclidean space, harmonic functions on these sets may be locally constant without being constant in their entire domain. In that case we say that the fractal has a degenerate harmonic structure. We prove that for a family of variants of the Sierpinski gasket the harmonic structure is non-degenerate. In paper III, we investigate properties of the Kusuoka measure and the corresponding energy Laplacian on the Sierpinski gaskets of level k. In papers IV and V, we establish a connection between the discrete combinatorial graph Laplacian determinant and the regularized determinant of the fractal itself. We establish that for a certain class of p.c.f. fractals the logarithm of the regularized determinant appears as a constant in the logarithm of the discrete combinatorial Laplacian