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UNIVERSITY of CALIFORNIA RIVERSIDE Spectral Zeta Functions UNIVERSITY OF CALIFORNIA RIVERSIDE Spectral Zeta Functions of Laplacians on Self-Similar Fractals A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Nishu Lal June 2012 Dissertation Committee: Professor Michel L. Lapidus, Chairperson Professor Fred Wilhelm Professor James Kelliher Copyright by Nishu Lal 2012 The Dissertation of Nishu Lal is approved: Committee Chairperson University of California, Riverside Acknowledgments It would not have been possible to write my dissertation thesis without the help and support of many kind and patient people. First and foremost, I would like to express my gratitude to my advisor, Dr. Michel Lapidus, who has been an inspirational mentor throughout this journey. I am very thankful to him for his unwavering support and guidance in the development of this research. I truly appreciate Dr. Lapidus for his continuous encouragement which helped me grow tremendously as a professional. I could not have asked for a better advisor. I would also like to thank my committee members, Dr. Fred Wilhelm and Dr. James Kelliher, for their time and effort in reading my thesis. I am also thankful for my research group members, Robert Niemeyer, Hafedh Herichi, Jonathan Sarhad, Michael Maroun, John Quinn, Scot Childress, and Vicente Alvarez, for providing interesting and insightful discussions on fractal geometry and mathematical physics. I would also like to thank the graduate division of the University of California, Riverside for the dissertation year fellowship that supported my work and the writing of my thesis. I would like to thank my parents, Neelam and Chaman, for their unconditional love and support. I appreciate my mom for the time she spent waking me up early in the morning when my alarm failed to go off, for her devotion for guiding me to aim higher and achieve my goals and for her endless sweet love and care which I will cherish for life. I cannot thank my dad enough for helping me realize that with determination and hard work pursuing my goal of higher education was not an impossible task. I would also like to thank my grandparents and my great-grandfather for giving me strength and heightening my confidence with their simple words of encouragement, \you can do it!" I am also thankful to my uncles, Vijay and Ashok, who posed as loving father figures iv and were always there when I was going through difficult times in my life. I would also like to thank my beautiful sisters, Neetu, Gagandeep, and Krustina and my wonderful brother, Palwinder, who were my cheerleaders, constantly encouraging and motivating me throughout this stressful and challenging journey. A very special thanks to Neetu for the late night motivating conversations and even cooking for me when taking a break from work seemed implausible. I cherish every moment with them and I am very blessed to have them in my life. Finally, I would like to thank my entire extended family for their continuous encouragement and support. v I dedicate my thesis to the loving memory of my mother, Neelam Lal. You will always remain very close to my heart. I miss you everyday, mom. vi ABSTRACT OF THE DISSERTATION Spectral Zeta Functions of Laplacians on Self-Similar Fractals by Nishu Lal Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, June 2012 Professor Michel L. Lapidus, Chairperson 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 renormaliza- tion 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 C (in the first case) or (in the second case) 2 as a polynomial on the complex projective plane, P (C). 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 factor- ization 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 measure on [0; 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain 2 polynomial in P (C), thereby extending to several variables an analogous result by A. Teplyaev. viii Contents List of Figures xi 1 Introduction 1 2 Background 5 2.1 Self-Similar Sets . .5 2.1.1 Hausdorff Metric . .5 2.1.2 Iterated Function System (IFS) . .6 2.1.3 Examples . .7 2.2 Laplacians on P.C.F. Fractals . .8 2.2.1 Self-Similar Measures . .9 2.2.2 Dirichlet Forms and Laplacians . 11 2.2.2.1 Weak Formulation . 11 2.2.2.2 The Pointwise Formulation . 12 2.2.3 Examples . 13 2.3 The Decimation Method . 15 2.4 Dirichlet Spaces . 19 2.4.1 Self-Adjoint Operators and Quadratic Forms . 19 2.5 Introduction To Hyperfunctions . 22 2.5.1 Motivation . 22 2.5.2 Definitions and Examples . 23 2.6 Complex Dynamics of Functions of Several Complex Variables . 27 3 The Spectral Zeta Function of the Laplacian on Fractals 30 3.1 Examples of Factorization of the Spectral Zeta Function . 31 3.1.1 The Spectral Zeta Function Associated with a Fractal String . 32 3.1.2 Cantor Fractal String . 34 3.2 The Spectral Zeta Function of the Laplacian on the Bounded Sierpinski Gasket . 35 3.3 Results for the Infinite (Unbounded) Sierpinski Gasket . 37 3.3.1 Main Lemma Regarding the Dirac Delta Hyperfunction . 39 3.4 A Representation of the Riemann Zeta Function . 42 4 Factorization of the Spectral Zeta Function of the Generalized Differ- ential Operators 44 4.1 The Fractal Sturm{Liouville Operator . 45 ix 4.2 The Eigenvalue Problem . 48 4.3 The Renormalization Map and the Spectrum of the Operator . 49 4.4 The Zeta Function Associated with the Renormalization Map . 54 4.5 The Sturm{Liouville Operator on the Half-Line and the Dirac Hyper- function . 56 1 4.6 The case α = 2 : Connection with the Riemann zeta function . 57 4.7 Concluding Comments . 59 Bibliography 61 x List of Figures 2.1 Approximations to the Cantor set . .7 2.2 Approximations to the Sierpinski gasket . .7 2.3 The Sierpinski gasket . .8 2.4 The eigenvalue diagram for SG . 18 3.1 An infinite Sierpinski gasket . 38 xi Chapter 1 Introduction In functional analysis, the second order differential equations describe many physical phenomena such as heat flow of a region over time with the assumption that the underlying space is smooth. As we come to learn, there are many objects in the world that are not so regular. How should one continue to carry out such analysis if the smoothness condition is removed? Considering the geometry of irregular objects, B. Mandelbrot introduced a 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 literature. We begin with the question of how to define the Lapla- cian on objects that are not smooth. The Sierpinski gasket is a commonly used fractal to study the construction of Laplacian on fractals. The first mathematical approach by S. Goldstein and S. Kusuoka [12, 20] introduced the Laplacian on the Sierpinski gasket as the generator of the diffusion process. Later on, J. Kigami [18] proposed a direct analytic approach using the theory of Dirichlet forms which was extended to a class of fractals called post-critically finite self-similar sets. 1 The physicists R. Rammal [32] and R. Rammal and G. Toulouse [33] studied the spectrum of the Laplacian of the Sierpinski gasket, in particular, the eigenvalue equation −∆µu = λu, and discovered the decimation method which establishes the relations between the spectrum of the Laplacian and the dynamics of the iteration of some polynomial on C. Later on, T. Shima [40] and M. Fukushima and T. Shima [11] gave a precise mathematical statement of their results. The development of the study of spectral zeta functions on fractals is inspired by M. Lapidus [22, 27] in light of the theory of fractal strings. A spectral zeta function is a zeta function that comes from the eigenvalue spectrum of a suitable differential operator. A fractal string L is a countable collection of disjoint intervals of lengths. He discovered a factorization of the spectral zeta function of the Dirichlet Laplacian L on a fractal string L of the following form −s ζL(s) = π ζ(s)ζL(s); (1.1) P1 s P1 −s where ζL(s) = j=1 `j is the geometric zeta function on L and ζ(s) = n=1 n is the Riemann zeta function. Later, A. Teplyaev [44, 45] studied the spectral zeta function of the Laplacian on a class of symmetric finitely ramified fractals extending the factorization for the fractal strings. He expressed the spectral zeta function of the Laplacian on the bounded Sierpinski gasket in terms of a new zeta function associated with the iterates of a polynomial in C.
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