Selberg Zeta Functions And Transfer Operators
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Selberg Zeta Functions And Transfer Operators
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Author : Markus Szymon Fraczek
language : en
Publisher: Springer
Release Date : 2017-05-11
Selberg Zeta Functions And Transfer Operators written by Markus Szymon Fraczek and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-05-11 with Mathematics categories.
This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non-singular perturbations. Areas in which the theory has not yet been sufficiently developed, such as the spectral theory of transfer operators or the singular perturbation theory of hyperbolic Laplacians, will profit from the numerical experiments discussed in this book. Detailed descriptions of numerical approaches to the spectra and eigenfunctions of transfer operators and to computations of Selberg zeta functions will be of value to researchers active in analysis, while those researchers focusing more on numerical aspects will benefit from discussions of the analytic theory, in particular those concerning the transfer operator method and the spectral theory of hyperbolic spaces.
Selberg Zeta And Theta Functions
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Author : Ulrich Bunke
language : en
Publisher: De Gruyter Akademie Forschung
Release Date : 1995
Selberg Zeta And Theta Functions written by Ulrich Bunke and has been published by De Gruyter Akademie Forschung this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
The authors give a self contained exposition of the theory of Selberg zeta and theta functions for bundles on compact locally symmetric spaces of rank 1. The connection between these functions and the spectrum of certain elliptic differential operators is provided by a version of the Selberg trace formula. The theta function is a regularized trace of the wave group. Originally defined geometrically, the Selberg zeta function has a representation in terms of regularized determinants. This leads to a complete description of its singularities. These results are employed in order to establish a functional equation and further properties of the Ruelle zeta function. A couple of explicit examples is worked out. Additional chapters are devoted to the theta function of Riemannian surfaces with cusps and to alternative descriptions of the singularities of the Selberg zeta function in terms of Lie algebra and group cohomology.
Character Deformation Of The Selberg Zeta Function For Congruence Subgroups Via The Transfer Operator
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Author : Markus Szymon Fraczek
language : en
Publisher:
Release Date : 2012
Character Deformation Of The Selberg Zeta Function For Congruence Subgroups Via The Transfer Operator written by Markus Szymon Fraczek and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
Dynamical Spectral And Arithmetic Zeta Functions
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Author : Michel Laurent Lapidus
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Dynamical Spectral And Arithmetic Zeta Functions written by Michel Laurent Lapidus and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.
Cohomological Theory Of Dynamical Zeta Functions
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Author : Andreas Juhl
language : en
Publisher: Springer Science & Business Media
Release Date : 2001
Cohomological Theory Of Dynamical Zeta Functions written by Andreas Juhl and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
The Lefschetz formula method can be considered as a link between the automorphic approach (Selberg trace formula) and Ruelle's approach (transfer operators). It yields a uniform cohomological characterization of the zeros and poles of the zeta functions and a new understanding of the functional equations from an index theoretical point of view. The divisors of the Selberg zeta functions also admit characterizations in terms of harmonic currents on the sphere bundle which represent the cohomology classes in the Lefschetz formulas in the sense of a Hodge theory. The concept of harmonic currents to be used for that purpose is introduced here for the first time.
Eigenfunctions Of Transfer Operators And Automorphic Forms For Hecke Triangle Groups Of Infinite Covolume
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Author : Roelof Bruggeman
language : en
Publisher: American Mathematical Society
Release Date : 2023-07-31
Eigenfunctions Of Transfer Operators And Automorphic Forms For Hecke Triangle Groups Of Infinite Covolume written by Roelof Bruggeman and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-31 with Mathematics categories.
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Cohomological Theory Of Dynamical Zeta Functions
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Author : Andreas Juhl
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Cohomological Theory Of Dynamical Zeta Functions written by Andreas Juhl and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Dynamical zeta functions are associated to dynamical systems with a countable set of periodic orbits. The dynamical zeta functions of the geodesic flow of lo cally symmetric spaces of rank one are known also as the generalized Selberg zeta functions. The present book is concerned with these zeta functions from a cohomological point of view. Originally, the Selberg zeta function appeared in the spectral theory of automorphic forms and were suggested by an analogy between Weil's explicit formula for the Riemann zeta function and Selberg's trace formula ([261]). The purpose of the cohomological theory is to understand the analytical properties of the zeta functions on the basis of suitable analogs of the Lefschetz fixed point formula in which periodic orbits of the geodesic flow take the place of fixed points. This approach is parallel to Weil's idea to analyze the zeta functions of pro jective algebraic varieties over finite fields on the basis of suitable versions of the Lefschetz fixed point formula. The Lefschetz formula formalism shows that the divisors of the rational Hassc-Wcil zeta functions are determined by the spectra of Frobenius operators on l-adic cohomology.
Dynamics And Numbers
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Author : Sergiǐ Kolyada:
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-07-27
Dynamics And Numbers written by Sergiǐ Kolyada: and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-07-27 with Ergodic theory categories.
This volume contains a collection of survey and research articles from the special program and international conference on Dynamics and Numbers held at the Max-Planck Institute for Mathematics in Bonn, Germany in 2014. The papers reflect the great diversity and depth of the interaction between number theory and dynamical systems and geometry in particular. Topics covered in this volume include symbolic dynamics, Bratelli diagrams, geometry of laminations, entropy, Nielsen theory, recurrence, topology of the moduli space of interval maps, and specification properties.
Hyperbolic Geometry And Applications In Quantum Chaos And Cosmology
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Author : Jens Bölte
language : en
Publisher: Cambridge University Press
Release Date : 2012
Hyperbolic Geometry And Applications In Quantum Chaos And Cosmology written by Jens Bölte and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Leading experts introduce this classical subject with exciting new applications in theoretical physics.
Emerging Applications Of Number Theory
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Author : Dennis A. Hejhal
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Emerging Applications Of Number Theory written by Dennis A. Hejhal and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Most people tend to view number theory as the very paradigm of pure mathematics. With the advent of computers, however, number theory has been finding an increasing number of applications in practical settings, such as in cryptography, random number generation, coding theory, and even concert hall acoustics. Yet other applications are still emerging - providing number theorists with some major new areas of opportunity. The 1996 IMA summer program on Emerging Applications of Number Theory was aimed at stimulating further work with some of these newest (and most attractive) applications. Concentration was on number theory's recent links with: (a) wave phenomena in quantum mechanics (more specifically, quantum chaos); and (b) graph theory (especially expander graphs and related spectral theory). This volume contains the contributed papers from that meeting and will be of interest to anyone intrigued by novel applications of modern number-theoretical techniques.