Spectral Theory on Fractals

Abstract

The main focus of this thesis is the study of Laplacians on fractals. Generalising the Laplace operator to domains that are not smooth is nothing new, Kigami and Krein respectively defined their operators and their properties were studied. Krein theory mainly deals with intervals with a Borel measure, whereas Kigami's Laplacian works with post-critically finite structures. The spectral theory of these operators has been actively studied. In particular, the asymptotic behaviour of the eigenvalue counting function under a given boundary condition. It remains true that for these generalised operators, the eigenvalue counting function obeys a power law as its asymptotic behaviour.

The goal of this thesis is threefold. We first show the equivalence between Kigami's Laplacian and the Krein operator when the underlying domain is the unit interval. Then, we derive an analytic function that encodes periodic eigenvalues on the circle as zeros of the function and the eigenvalue multiplicity as the zero multiplicity.

Second, we study eigenfunctions on the Sierpiński gasket and proved a method to construct eigenfunctions for Kigami's Laplacian. This method works on measures that are not just the uniform distribution, covering more cases than the spectral decimation method first proposed by Fukishima and Shima in 1992.

Finally, we generalise the notion of a post-critically finite (PCF) structure from self-similar structures to generalised graph directed Markov systems (GGDMS), enabling a more general version of Kigami's Laplacian to be defined. We then prove that the spectral exponent of Laplacians may be expressed in terms of a dimension value that is reminiscent of the Minkowski dimension if the energy form satisfies the bounded-distortion property. Furthermore, if the energy form is generated from a Hölder potential function and the underlying measure is a Gibbs measure, then the spectral exponent is the unique zero of a pressure formula. This result can be seen as a generalisation to Kesseböhmer and Niemann's work in 2022, where the pressure formula they derived is done in the context of Krein strings. We end the thesis by giving a few examples that illustrate the framework under GGDMS.

Date of Award	19 Mar 2024
Original language	English
Awarding Institution	University of Bristol
Supervisor	Thomas M Jordan (Supervisor) & Asma Hassannezhad (Supervisor)