Högskolan i Skövde

Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem 3.4. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski gasket, a non-post critically finite analog of the Sierpinski gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.

We prove that the harmonic extension matrices for the two dimensional level-k Sierpiński gasket are invertible for every k ≥ 2. This has been previously conjectured to be true by Hino in [10] and [11] and tested numerically for k ≤ 50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.

We extend and survey results in the theory of analysis on fractal sets from the standard Laplacian on the Sierpinski gasket to the energy Laplacian, which is defined weakly by using the Kusuoka energy measure. We also extend results from the Sierpinski gasket to level-k Sierpinski gaskets, for all k ≤ 2. We observe that the pointwise formula for the energy Laplacian is valid for all level-k Sierpinski gaskets, SGk, and we provide a proof of a known formula for the renormalization constants of the Dirichlet form for postcritically finite self-similar sets along with a probabilistic interpretation of the Laplacian pointwise formula. We also provide a vector self-similar formula and a variable weight self-similar formula for the Kusuoka measure on SGk, as well as a formula for the scaling of the energy Laplacian. Copyright © 2019 Rocky Mountain Mathematics Consortium.

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.