Dynamical Zeta Functions And Dynamical Determinants For Hyperbolic Maps

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Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

Viviane Baladi
Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps Book PDF

Author : Viviane Baladi
Publisher : Springer
Page : 291 pages
File Size : 54,6 Mb
Release : 2018-05-09
Category : Mathematics
ISBN : 9783319776613

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Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps by Viviane Baladi Pdf

The spectra of transfer operators associated to dynamical systems, when acting on suitable Banach spaces, contain key information about the ergodic properties of the systems. Focusing on expanding and hyperbolic maps, this book gives a self-contained account on the relation between zeroes of dynamical determinants, poles of dynamical zeta functions, and the discrete spectra of the transfer operators. In the hyperbolic case, the first key step consists in constructing a suitable Banach space of anisotropic distributions. The first part of the book is devoted to the easier case of expanding endomorphisms, showing how the (isotropic) function spaces relevant there can be studied via Paley–Littlewood decompositions, and allowing easier access to the construction of the anisotropic spaces which is performed in the second part. This is the first book describing the use of anisotropic spaces in dynamics. Aimed at researchers and graduate students, it presents results and techniques developed since the beginning of the twenty-first century.

Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval

David Ruelle
Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval Book PDF

Author : David Ruelle
Publisher : American Mathematical Soc.
Page : 74 pages
File Size : 43,6 Mb
Release : 1994
Category : Mathematics
ISBN : 0821836013

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Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval by David Ruelle Pdf

With a general introduction to the subject, this title presents a detailed study of the zeta functions associated with piecewise monotone maps of the interval $ 0,1]$. In particular, it gives a proof of a generalized form of the Baladi-Keller theorem relating the poles of $\zeta (z)$ and the eigenvalues of the transfer operator.

Dynamical, Spectral, and Arithmetic Zeta Functions

Michel Laurent Lapidus,Spectral AMS Special Session on Dynamical,Machiel Van Frankenhuysen
Dynamical Spectral and Arithmetic Zeta Functions Book PDF

Author : Michel Laurent Lapidus,Spectral AMS Special Session on Dynamical,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 210 pages
File Size : 41,8 Mb
Release : 2001
Category : Functions, Zeta
ISBN : 9780821820797

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Dynamical, Spectral, and Arithmetic Zeta Functions by Michel Laurent Lapidus,Spectral AMS Special Session on Dynamical,Machiel Van Frankenhuysen Pdf

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.

Thermodynamic Formalism

Mark Pollicott,Sandro Vaienti
Thermodynamic Formalism Book PDF

Author : Mark Pollicott,Sandro Vaienti
Publisher : Springer Nature
Page : 536 pages
File Size : 44,9 Mb
Release : 2021-10-01
Category : Mathematics
ISBN : 9783030748630

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Thermodynamic Formalism by Mark Pollicott,Sandro Vaienti Pdf

This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.

Cohomological Theory of Dynamical Zeta Functions

Andreas Juhl
Cohomological Theory of Dynamical Zeta Functions Book PDF

Author : Andreas Juhl
Publisher : Birkhäuser
Page : 712 pages
File Size : 48,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883405

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Cohomological Theory of Dynamical Zeta Functions by Andreas Juhl Pdf

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.

Positive Transfer Operators And Decay Of Correlations

Viviane Baladi
Positive Transfer Operators And Decay Of Correlations Book PDF

Author : Viviane Baladi
Publisher : World Scientific
Page : 326 pages
File Size : 53,6 Mb
Release : 2000-07-12
Category : Science
ISBN : 9789814496667

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Positive Transfer Operators And Decay Of Correlations by Viviane Baladi Pdf

Although individual orbits of chaotic dynamical systems are by definition unpredictable, the average behavior of typical trajectories can often be given a precise statistical description. Indeed, there often exist ergodic invariant measures with special additional features. For a given invariant measure, and a class of observables, the correlation functions tell whether (and how fast) the system “mixes”, i.e. “forgets” its initial conditions.This book, addressed to mathematicians and mathematical (or mathematically inclined) physicists, shows how the powerful technology of transfer operators, imported from statistical physics, has been used recently to construct relevant invariant measures, and to study the speed of decay of their correlation functions, for many chaotic systems. Links with dynamical zeta functions are explained.The book is intended for graduate students or researchers entering the field, and the technical prerequisites have been kept to a minimum.

Classical Nonintegrability, Quantum Chaos

Andreas Knauf,Yakov G. Sinai
Classical Nonintegrability Quantum Chaos Book PDF

Author : Andreas Knauf,Yakov G. Sinai
Publisher : Birkhäuser
Page : 104 pages
File Size : 40,7 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034889322

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Classical Nonintegrability, Quantum Chaos by Andreas Knauf,Yakov G. Sinai Pdf

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator. Similar to the use made of the Riemann zeta function in the investigation of the prime numbers, dynamical zeta functions are now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first introduces dynamical zeta functions and transfer operators, illustrating and motivating these notions with a simple one-dimensional dynamical system. Then she presents a commented list of useful references, helping the newcomer to enter smoothly into this fast-developing field of research. Chapter 3 on irregular scattering and Chapter 4 on quantum chaos by A. Knauf deal with solutions of the Hamilton and the Schr6dinger equation. Scatter ing by a potential force tends to be irregular if three or more scattering centres are present, and a typical phenomenon is the occurrence of a Cantor set of bounded orbits. The presence of this set influences those scattering orbits which come close.

Geometric and Probabilistic Structures in Dynamics

Workshop on Dynamical Systems and Related Topics
Geometric and Probabilistic Structures in Dynamics Book PDF

Author : Workshop on Dynamical Systems and Related Topics
Publisher : American Mathematical Soc.
Page : 358 pages
File Size : 55,7 Mb
Release : 2008
Category : Differentiable dynamical systems
ISBN : 9780821842867

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Geometric and Probabilistic Structures in Dynamics by Workshop on Dynamical Systems and Related Topics Pdf

"This book presents a collection of articles that cover areas of mathematics related to dynamical systems. The authors are well-known experts who use geometric and probabilistic methods to study interesting problems in the theory of dynamical systems and its applications. Some of the articles are surveys while others are original contributions. The topics covered include: Riemannian geometry, models in mathematical physics and mathematical biology, symbolic dynamics, random and stochastic dynamics. This book can be used by graduate students and researchers in dynamical systems and its applications."--BOOK JACKET.

Dynamical Zeta Functions, Nielsen Theory, and Reidemeister Torsion

Alexander Fel'shtyn
Dynamical Zeta Functions Nielsen Theory and Reidemeister Torsion Book PDF

Author : Alexander Fel'shtyn
Publisher : American Mathematical Soc.
Page : 170 pages
File Size : 54,6 Mb
Release : 2000-01-01
Category : Mathematics
ISBN : 0821864211

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Dynamical Zeta Functions, Nielsen Theory, and Reidemeister Torsion by Alexander Fel'shtyn Pdf

In the paper we study new dynamical zeta functions connected with Nielsen fixed point theory. The study of dynamical zeta functions is part of the theory of dynamical systems, but it is also intimately related to algebraic geometry, number theory, topology and statistical mechanics. The paper consists of four parts. Part I presents a brief account of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with analog of Dold congruences for the Reidemeister and Nielsen numbers. In Part IV we explain how dynamical zeta functions give rise to the Reidemeister torsion, a very important topological invariant which has useful applications in knots theory,quantum field theory and dynamical systems.

Handbook of Dynamical Systems

A. Katok,B. Hasselblatt
Handbook of Dynamical Systems Book PDF

Author : A. Katok,B. Hasselblatt
Publisher : Elsevier
Page : 1235 pages
File Size : 55,7 Mb
Release : 2005-12-17
Category : Mathematics
ISBN : 9780080478227

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Handbook of Dynamical Systems by A. Katok,B. Hasselblatt Pdf

This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey “Principal Structures of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field.. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.

Smooth Ergodic Theory and Its Applications

A. B. Katok
Smooth Ergodic Theory and Its Applications Book PDF

Author : A. B. Katok
Publisher : American Mathematical Soc.
Page : 895 pages
File Size : 44,9 Mb
Release : 2001
Category : Ergodic theory
ISBN : 9780821826829

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Smooth Ergodic Theory and Its Applications by A. B. Katok Pdf

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student-or even an established mathematician who is not an expert in the area-to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference. Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincare and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of corre This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

Handbook of Dynamical Systems

B. Hasselblatt,A. Katok
Handbook of Dynamical Systems Book PDF

Author : B. Hasselblatt,A. Katok
Publisher : Elsevier
Page : 1232 pages
File Size : 48,6 Mb
Release : 2002-08-20
Category : Mathematics
ISBN : 9780080533445

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Handbook of Dynamical Systems by B. Hasselblatt,A. Katok Pdf

Volumes 1A and 1B. These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys. The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics. Volume 1B will appear 2005.

European Congress of Mathematics

Carles Casacuberta,Rosa Maria Miro-Roig,Joan Verdera,Sebastia Xambo-Descamps
European Congress of Mathematics Book PDF

Author : Carles Casacuberta,Rosa Maria Miro-Roig,Joan Verdera,Sebastia Xambo-Descamps
Publisher : Birkhäuser
Page : 611 pages
File Size : 47,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034882682

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European Congress of Mathematics by Carles Casacuberta,Rosa Maria Miro-Roig,Joan Verdera,Sebastia Xambo-Descamps Pdf

This is the first volume of the proceedings of the third European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners as well as papers by plenary and parallel speakers. The second volume collects articles by prize winners and speakers of the mini-symposia. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: R. Ahlswede, V. Bach, V. Baladi, J. Bruna, N. Burq, X. Cabré, P.J. Cameron, Z. Chatzidakis, C. Ciliberto, G. Dal Maso, J. Denef, R. Dijkgraaf, B. Fantechi, H. Föllmer, A.B. Goncharov, A. Grigor'yan, M. Harris, R. Iturriaga, K. Johansson, K. Khanin, P. Koskela, H.W. Lenstra, Jr., F. Loeser, Y.I. Manin, N.S. Manton, Y. Meyer, I. Moerdijk, E.M. Opdam, T. Peternell, B.M.A.G. Piette, A. Reznikov, H. Schlichtkrull, B. Schmidt, K. Schmidt, C. Simó, B. Tóth, E. van den Ban, M.-F. Vignéras, O. Viro.

Arithmetic L-Functions and Differential Geometric Methods

Pierre Charollois,Gerard Freixas i Montplet,Vincent Maillot
Arithmetic L Functions and Differential Geometric Methods Book PDF

Author : Pierre Charollois,Gerard Freixas i Montplet,Vincent Maillot
Publisher : Springer Nature
Page : 324 pages
File Size : 48,5 Mb
Release : 2021-05-17
Category : Mathematics
ISBN : 9783030652036

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Arithmetic L-Functions and Differential Geometric Methods by Pierre Charollois,Gerard Freixas i Montplet,Vincent Maillot Pdf

This book is an outgrowth of the conference “Regulators IV: An International Conference on Arithmetic L-functions and Differential Geometric Methods” that was held in Paris in May 2016. Gathering contributions by leading experts in the field ranging from original surveys to pure research articles, this volume provides comprehensive coverage of the front most developments in the field of regulator maps. Key topics covered are: • Additive polylogarithms • Analytic torsions • Chabauty-Kim theory • Local Grothendieck-Riemann-Roch theorems • Periods • Syntomic regulator The book contains contributions by M. Asakura, J. Balakrishnan, A. Besser, A. Best, F. Bianchi, O. Gregory, A. Langer, B. Lawrence, X. Ma, S. Müller, N. Otsubo, J. Raimbault, W. Raskin, D. Rössler, S. Shen, N. Triantafi llou, S. Ünver and J. Vonk.

Modeling, Dynamics, Optimization and Bioeconomics I

Alberto Adrego Pinto,David Zilberman
Modeling Dynamics Optimization and Bioeconomics I Book PDF

Author : Alberto Adrego Pinto,David Zilberman
Publisher : Springer
Page : 753 pages
File Size : 43,8 Mb
Release : 2014-06-20
Category : Mathematics
ISBN : 9783319048499

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Modeling, Dynamics, Optimization and Bioeconomics I by Alberto Adrego Pinto,David Zilberman Pdf

This volume explores the emerging and current, cutting-edge theories and methods of modeling, optimization, dynamics and bio economy. It provides an overview of the main issues, results and open questions in these fields as well as covers applications to biology, economy, energy, industry, physics, psychology and finance. The majority of the contributed papers for this volume come from the participants of the International Conference on Modeling, Optimization and Dynamics (ICMOD 2010), a satellite conference of EURO XXIV Lisbon 2010, which took place at Faculty of Sciences of University of Porto, Portugal and from the Berkeley Bio economy Conference 2012, at the University of California, Berkeley, USA.