Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

A Functional Approach
Author: Viviane Baladi
Publisher: Springer
ISBN: 3319776614
Category: Mathematics
Page: 291
View: 5569

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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.

Positive Transfer Operators and Decay of Correlations


Author: Viviane Baladi
Publisher: World Scientific
ISBN: 9814496669
Category: Science
Page: 324
View: 321

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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. Contents:Subshifts of Finite Type: A Key Symbolic ModelSmooth Uniformly Expanding DynamicsPiecewise Expanding SystemsHyperbolic Systems Readership: Physicists, mathematicians, interdisciplinary scientists and social scientists. Keywords:Transfer Operator;Spectrum;Thermodynamic Formalism;Correlation Function;Mixing;Dynamical System;Chaos;Equilibrium State;Spectral Gap;Perron Frobenius OperatorReviews: “The book can be used both as a textbook for beginners and as a reference book for workers in the field … the proofs are given in full, allowing a detailed study of the techniques and making the book nicely self-contained.” Mathematical Reviews

Cohomological Theory of Dynamical Zeta Functions


Author: Andreas Juhl
Publisher: Birkhäuser
ISBN: 3034883404
Category: Mathematics
Page: 709
View: 4004

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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.

European Congress of Mathematics, Barcelona, July 10-14, 2000

Vol. 1
Author: Spain) European Congress of Mathematics 2000 (Barcelona
Publisher: Springer Science & Business Media
ISBN: 9783764364175
Category: Mathematics
Page: 582
View: 4229

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The Third European Congress of Mathematics (3ecm) took place from July 10th to July 14th, 2000 in Barcelona. It was organised by the Societat Catalana de Matematiques (Catalan Mathematical Society), under the auspices of the Euro­ pean Mathematical Society (EMS). As foreseen by the EMS and the International Mathematical Union, this Congress was a major event in World Mathematical Year 2000. In addition to reviewing outstanding research achievements, important aspects of the life of European mathematics were discussed. Mathematics is undergoing a period of rapid changes. Effective computation and applications in science and technology go ever more hand in hand with con­ ceptual developments. It was one of the aims of 3ecm to reflect this mutual enrich­ ment, while steering present and future trends of mathematical sciences. In fact, the motto of the Congress, Shaping the 21st Century, was meant to capture these views. Nearly 1400 people from 87 countries gathered together in the Palau de Con­ gressos of Barcelona in order to take part in the activities of the 3ecm scientific programme: Nine plenary lectures, thirty invited lectures in parallel sessions, lec­ tures given by EMS prize winners, ten mini-symposia on special topics, seven round tables, poster sessions, presentations of mathematical software and video exhibitions. Twenty events were satellites of 3ecm in various countries.

Spectral Theory of Infinite-Area Hyperbolic Surfaces


Author: David Borthwick
Publisher: Springer Science & Business Media
ISBN: 0817645241
Category: Mathematics
Page: 355
View: 7704

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By focusing on the scattering theory of hyperbolic surfaces, this work provides an introduction to the geometry of hyperbolic surfaces. Aimed at graduate students and researchers, it draws on techniques from functional analysis and differential geometry, as well as some techniques from algebra and number theory.

Harmonic Analysis and Rational Approximation

Their Rôles in Signals, Control and Dynamical Systems
Author: Jean-Daniel Fournier
Publisher: Springer Verlag
ISBN: 9783540309222
Category: Science
Page: 311
View: 5582

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Research in Interactive Design presents an in-depth study of new product design methods. Interactive design is an original structured design approach that has recently emerged from transversal research. On the research side, the topic of interactive design is the fruitful result of sophisticated joint studies combining specific numerical engineering theories, original development in design science and industrial engineering, new discoveries in mechatronics and advanced mechanical engineering methods. Additionally, industry benefits from interactive design instruments. Interactive design is a constructive approach that tends to ensure innovation by improving user-integration in the design process, fostering the development of virtual prototypes and providing new powerful ways for collaborative design activities. Interactive design instruments are seen as the extension of traditional CAD tools. This book provides an accurate overview of the recent research or industrial application in interactive design. The different arguments, taken from the international conference Virtual Concept 2005, will provide the reader with some advanced solutions concerning new methods and tools by discussing: - modelling techniques: new techniques ensuring the development of high qualified, updated and reduced models being able to represent the behaviour of a process or a product in its environment; - design solution space exploration: specific instruments dedicated to the identification of suitable design solutions and their virtual representation allowing engineers to have interaction with the virtual worlds; - interactive process organization: studies highlighting the basic evolutions of a process integrating new instruments of interactive design. Each subsequent part of the book addresses both advanced techniques and industrial applications of the latest successful research in interactive design. The solutions presented are enhanced by demonstration

Classical Nonintegrability, Quantum Chaos


Author: Andreas Knauf,Yakov G. Sinai
Publisher: Birkhäuser
ISBN: 3034889321
Category: Science
Page: 102
View: 5860

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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.

Frontiers in Number Theory, Physics, and Geometry I

On Random Matrices, Zeta Functions, and Dynamical Systems
Author: Pierre E. Cartier,Bernard Julia,Pierre Moussa,Pierre Vanhove
Publisher: Springer
ISBN: 9783540231899
Category: Mathematics
Page: 624
View: 2441

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The relation between mathematics and physics has a long history, in which the role of number theory and of other more abstract parts of mathematics has recently become more prominent. More than 10 years after a first meeting between number theorists and physicists at the Centre de Physique des Houches, a second two-week event focused on the broader interface of number theory, geometry, and physics. This book collects the material presented at this meeting.

Einführung in die Struktur- und Darstellungstheorie der klassischen Gruppen


Author: Wolfgang Hein
Publisher: Springer-Verlag
ISBN: 3642743404
Category: Mathematics
Page: 255
View: 6408

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Eine gleichermaßen aktuelle wie zusammenfassende Darstellung der wichtigsten Methoden zur Untersuchung der klassischen Gruppen fehlte bislang in deutschsprachigen Lehrbüchern. Indem der Autor die klassischen Gruppen sowohl aus algebraisch-geometrischer Sicht, wie auch mit Lieschen (infinitesimalen) Methoden studiert, schließt er diese Lücke. Die von Grund auf behandelte Darstellungstheorie mündet im algebraischen Teil in der Brauer-Weylschen Methode der Zerlegung von Tensorpotenzen durch Youngsche Symmetrieoperatoren in irreduzible Teilräume. Auf der Ebene der Lie-Algebren wird die Klassifikation der irreduziblen Darstellungen durch höchste Gewichte durchgeführt. Besonderer Wert liegt auf einer ausführlichen Erläuterung des Zusammenspiels der Gruppen und ihrer Lie-Algebren, die das Kernstück der Lieschen Theorie ausmachen. In dieser Hinsicht dient das Buch auch als Einführung in die Theorie der Lie-Gruppen; zur Parametrisierung wird dabei ausschließlich die Matrix-Exponentialabbildung verwandt, wodurch ganz auf den aufwendigen Apparat der differenzierbaren Mannigfaltigkeiten verzichtet werden kann. Eine Fülle von Beispielen und Übungsaufgaben dienen zur Vertiefung des Gelernten. Inhaltlich schließt der Text unmittelbar an die Grundvorlesungen über Analysis und Lineare Algebra an.

Handbook of chaos control


Author: Heinz Georg Schuster
Publisher: Wiley-VCH Verlag GmbH
ISBN: 9783527294367
Category: Science
Page: 691
View: 2506

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Chaos, that is irregular dynamical behaviour, is ubiquituous in nature and occurs in a wide range of systems including lasers, fluids, etc., heart beats and brain waves. Before 1990 the emergence of chaos in a system was mostly considered as a nuisance because chaotic systems are hard to predict due to their sensitivity to small perturbations. After 1990 it became clear that this sensitive dependence offers the unique possibility to control these systems with a minimum of additional energy. This handbook provides a comprehensive up-to-date overview of the field. It starts with an introduction to chaos theory, and covers all known methods of chaos control from parametric feedback to neuronal networks. A large part of the handbook is devoted to applications which range from control of electronic circuits, the control of lasers and chemical plants up to "antichaos control" in biological systems which offers the possibility to avoid epileptic seizures.

Quantum Chaos

Between Order and Disorder
Author: Giulio Casati,Boris Chirikov
Publisher: Cambridge University Press
ISBN: 9780521031660
Category: Mathematics
Page: 700
View: 5992

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This book provides a comprehensive overview of our understanding of chaotic behaviour in quantum systems.

Vorlesungen über Atommechanik


Author: M. Born
Publisher: Springer-Verlag
ISBN: 3642618987
Category: Science
Page: 358
View: 6162

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Physical Review

Statistical physics, plasmas, fluids, and related interdisciplinary topics
Author: N.A
Publisher: N.A
ISBN: N.A
Category: Statistical physics
Page: N.A
View: 7704

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