Date of Completion

4-26-2017

Embargo Period

4-25-2017

Keywords

fractal, Julia set, self-similarity, Laplacian, eigenvalues, eigenfunctions, spectral decimation

Major Advisor

Alexander Teplyaev

Co-Major Advisor

Luke Rogers

Associate Advisor

Maria Gordina

Associate Advisor

see above

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

The Hata tree is the unique self-similar set in the complex plane determined by the contractions φ0(z) = cz and φ1(z) = (1-|c|2)z+|c|2, where c is a complex number such that |c| and |1-c| are in (0,1). There are four main results in the paper. First, by applying linear algebra and spectral theory it is possible to construct a dynamical system that can compute the eigenvalues of the probabilistic Laplacian on graph approximations to the Hata tree. Conclusions are made about the spectrum of the Laplacian on the limiting graphs. Second, the Sabot theory (c.f. [29]) is applied to construct a simpler dynamical system to compute the eigenvalues of a class of normalized graph Laplacians (including the probabilistic Laplacian) on these approximating graphs. Third, it is possible to reconstruct the Hata tree as the union of two copies of a mixed affine nested fractal identified at a point. Using techniques from [13], some results are stated on the spectral asymptotics of the eigenvalue counting function of a certain class of Laplacians (not including the probabilistic Laplacian) on this mixed affine nested fractal. In the final part, a spectral analysis is performed on graph approximations to the Basilica Julia set of the polynomial z2-1. In [5], the authors give a dynamical system that can be used to construct finite approximations and classify the different possible infinite blow-ups. In this paper, the techniques from the first part are used to construct a dynamical system that can compute the eigenvalues of Laplacian operators on these finite graph approximations. In addition, it is shown that the spectrum of the Laplacian on blow-ups satisfying certain conditions is pure point.

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