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α-Scaling Zeta Functions For Self-Similar Multifractals


Visualization of sets in Euclidean space that possess notions of non-integer dimension has lead to a great deal of curiosity and ideas that connect to many traditional fields in and outside of mathematics. Recently, the study of open subsets of R and their boundaries paved the way for a new definition in the work of Lapidus and van Frankenhuijsen for the term `fractal.' This definition reveals the scaling properties of the relevant sets and serves as continuing motivation for the study of zeta functions that encode a notion of dimension of a corresponding set. There are, in addition, measures called multifractals that possess fractal-like properties. It is the purpose of this dissertation to investigate the properties of self-similar multifractal measures through a very intimate connection to the classically studied hypergeometric functions. One may decompose the support of these measures into a collection of fractal sets each of which possesses the same scaling property. Zeta functions associated to each fractal set in this decomposition encode this generalized notion of dimension, yet they are shown in many cases to be exactly of the form of the generalized hypergeometric fuctions. The classical study of hypergeometric functions boasts the attention of mathematicians such as Euler, Gauss, and Riemann, yet as with many special functions the applications continue to be revealed. Some invaluable tools in understanding the generalized hypergeometric functions, as investigated herein, are the associated hypergeometric differential equation and monodromy matrices.

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