On the asymptotic distribution of the Dirichlet eigenvalues of fractal chains

In this thesis, we offer an investigation of the vibrational properties of discrete one-dimensional systems with an underlying fractal structure. Thus, the primary objects of scrutiny in this work are two types of fractal objects: the first class being quite simple structures with a fractal boundary, the second class having an internal fractal structure but very simple boundaries. By introducing a matrix representation of the related Laplacians, we prove the efficiency of using techniques originally taken from random matrix theory in the area of fractal geometry. Thereby, a unifying framework for the study of these systems has been developed, capable of being extended to higher dimensions.