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Author
Title Spectral Zeta Functions of Laplacians on Self-Similar Fractals
URL
Publication Date
Discipline/Department Mathematics
University/Publisher University of California – Riverside
Abstract This thesis investigates the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e., infinite) Sierpinski gasket and a self-similar Sturm – Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, C. Sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables, called the renormalization map. We obtain a factorization of the spectral zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on the complex plane (in the first case) or (in the second case) as a polynomial on the complex projective plane. Our first main result extends to the case of the fractal Laplacian on the unbounded Sierpinski gasket a factorization formula obtained by M. Lapidus for the spectral zeta function of a fractal string and later extended by A. Teplyaev to the bounded (i.e., finite) Sierpinski gasket and some other decimable fractals. Furthermore, our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm – Liouville operators. Moreover, as a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial on the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev.
Subjects/Keywords Mathematics; Analysis on fractals; decimation method; Dirac delta hyperfunction; fractal Sturm-Liouville operators; multivariable complex dynamics; spectral zeta functions
Language en
Rights public
Country of Publication us
Format application/pdf
Record ID california:qt888903d2
Other Identifiers qt888903d2
Repository california
Date Indexed 2018-02-26

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operators, a class of generalized second order differential operators of the form 2 d d dm dx . First, we show that the spectral zeta function of the Laplacian of the unbounded Sierpinski gasket can be written as the product of the Dirac delta…

…This thesis investigates the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e., infinite) Sierpinski gasket and a self-similar Sturm–Liouville operator associated with a fractal self-similar…

…zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on C…

…these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm–Liouville operators. Moreover, as a corollary, in the very special vii case when the underlying self-similar measure is Lebesgue…

…Dirichlet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Self-Adjoint Operators and Quadratic Forms . . . . . . 2.5 Introduction To Hyperfunctions . . . . . . . . . . . . . . . . . . 2.5.1 Motivation…

…Main Lemma Regarding the Dirac Delta Hyperfunction . . . . . 3.4 A Representation of the Riemann Zeta Function . . . . . . . . . . . . . 5 5 5 6 7 8 9 11 11 12 13 15 19 19 22 22 23 27 30 31 32 34 35 37 39 42 4 Factorization of the Spectral Zeta…

…Function of the Generalized Differential Operators 44 4.1 The Fractal Sturm–Liouville Operator . . . . . . . . . . . . . . . . . . . 45 ix 4.2 4.3 4.4 4.5 4.6 4.7 The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . The…

…Renormalization Map and the Spectrum of the Operator . . . . . . The Zeta Function Associated with the Renormalization Map . . . . . . The Sturm–Liouville Operator on the Half-Line and the Dirac Hyperfunction…

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