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Lal, Nishu.
Spectral Zeta Functions of Laplacians on Self-Similar * Fractals*.

Degree: Mathematics, 2012, University of California – Riverside

URL: http://www.escholarship.org/uc/item/888903d2

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

…mathematical framework for such objects, now known as the
theory of *analysis* *on* *fractals*.
The… …spectral *analysis* *on* *fractals* was first studied by R. Rammal and G.
Toulouse in physics… …2.2 Laplacians *on* P.C.F. *Fractals* . . . . . . . . . . . . . . . . . . .
2.2.1 Self-Similar… …3 The Spectral Zeta Function of the Laplacian *on* *Fractals*
3.1 Examples of Factorization of… …*on* *fractals*. The first mathematical approach by
S. Goldstein and S. Kusuoka [12, 20…

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APA (6^{th} Edition):

Lal, N. (2012). Spectral Zeta Functions of Laplacians on Self-Similar Fractals. (Thesis). University of California – Riverside. Retrieved from http://www.escholarship.org/uc/item/888903d2

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Lal, Nishu. “Spectral Zeta Functions of Laplacians on Self-Similar Fractals.” 2012. Thesis, University of California – Riverside. Accessed January 26, 2021. http://www.escholarship.org/uc/item/888903d2.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Lal, Nishu. “Spectral Zeta Functions of Laplacians on Self-Similar Fractals.” 2012. Web. 26 Jan 2021.

Vancouver:

Lal N. Spectral Zeta Functions of Laplacians on Self-Similar Fractals. [Internet] [Thesis]. University of California – Riverside; 2012. [cited 2021 Jan 26]. Available from: http://www.escholarship.org/uc/item/888903d2.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Lal N. Spectral Zeta Functions of Laplacians on Self-Similar Fractals. [Thesis]. University of California – Riverside; 2012. Available from: http://www.escholarship.org/uc/item/888903d2

Not specified: Masters Thesis or Doctoral Dissertation