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Author: Gabriel P. Paternain
Publisher: Springer Science & Business Media
ISBN: 1461216001
Category : Mathematics
Languages : en
Pages : 160
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Book Description
The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mane's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who helped me with the writing of the first two chapters and the figures. Gonzalo Tomaria caught several errors and contributed with helpful suggestions. Pablo Spallanzani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.
Author: Gabriel P. Paternain
Publisher: Springer Science & Business Media
ISBN: 1461216001
Category : Mathematics
Languages : en
Pages : 160
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Book Description
The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mane's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who helped me with the writing of the first two chapters and the figures. Gonzalo Tomaria caught several errors and contributed with helpful suggestions. Pablo Spallanzani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.
Author: Gabriel P. Paternain
Publisher: Springer Science & Business Media
ISBN: 9780817641443
Category : Mathematics
Languages : en
Pages : 172
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Book Description
"This self-contained monograph will be of interest to graduate students and researchers of dynamical systems and differential geometry. Numerous exercises and examples as well as a comprehensive index and bibliography make this work an excellent self-study resource or text for a one-semester course or seminar."--BOOK JACKET.
Author: Igor Nikolaev
Publisher: Springer Science & Business Media
ISBN: 9783540660804
Category : Mathematics
Languages : en
Pages : 324
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Book Description
Time-evolution in low-dimensional topological spaces is a subject of puzzling vitality. This book is a state-of-the-art account, covering classical and new results. The volume comprises Poincaré-Bendixson, local and Morse-Smale theories, as well as a carefully written chapter on the invariants of surface flows. Of particular interest are chapters on the Anosov-Weil problem, C*-algebras and non-compact surfaces. The book invites graduate students and non-specialists to a fascinating realm of research. It is a valuable source of reference to the specialists.
Author: Calvin C. Moore
Publisher: Springer Science & Business Media
ISBN: 1461247225
Category : Mathematics
Languages : en
Pages : 283
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Book Description
The Mathematical Sciences Research Institute sponsored a three day conference, May 21-23, 1984 to honor Professor George W. Mackey. The title of the conference, Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics, reflects the interests in science that have characterized Professor wide ranging Mackey's work. The conference provided an opportunity for his students, friends and colleagues to honor him and his contributions. The conference was attended by over one hundred people and the participants included five mathematical generations Professor Mackey's mathematical father, Marshall Stone, many mathematical children, grandchildren, and at least one mathematical great-grandchild. This volume is a compendium of the scientific papers presented at the conference plus some additional papers contributed after the conference. The far ranging scope of the various articles is a further indication of the large number of fields that have been affected by Professor Mackey's work. Calvin C. Moore Berkeley, CA Feb, 1986 Table of Contents Preface vi i Ambiguity Functions and Group L. Auslander and Representations R. Tolimieri Kirillov Orbits and Direct Integral Lawrence Corwin 11 Decompositions on Certain Quotient Spaces Some Homotopy and Shape Calculations Edward G. Effors and 69 for C*-Algebras Jerome Kaminker 121 Small Unitary Representations of Roger Howe Classical Groups Dual Vector Spaces Irving Kaplansky 151 Exponential Decay of Correlation Calvin C. Moore 163 Coefficients for Geodesic Flows Lattices in U(n. I) G. D. Mostow Induced Bundles and Nonlinear Irving E. Segal 199 Wave equations Compact Ahelian Aut.
Author:
Publisher: World Scientific
ISBN:
Category :
Languages : en
Pages : 1001
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Book Description
Author: Werner Ballmann
Publisher: Birkhäuser
ISBN: 3034892403
Category : Mathematics
Languages : en
Pages : 114
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Book Description
Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.
Author: Anatole Katok
Publisher: Cambridge University Press
ISBN: 9780521575577
Category : Mathematics
Languages : en
Pages : 828
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Book Description
This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.
Author: D. V. Anosov
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 442
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Book Description
Author: Luís Barreira
Publisher: Springer Science & Business Media
ISBN: 3319005480
Category : Mathematics
Languages : en
Pages : 155
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Book Description
The dimension theory of dynamical systems has progressively developed, especially over the last two decades, into an independent and extremely active field of research. Its main aim is to study the complexity of sets and measures that are invariant under the dynamics. In particular, it is essential to characterizing chaotic strange attractors. To date, some parts of the theory have either only been outlined, because they can be reduced to the case of maps, or are too technical for a wider audience. In this respect, the present monograph is intended to provide a comprehensive guide. Moreover, the text is self-contained and with the exception of some basic results in Chapters 3 and 4, all the results in the book include detailed proofs. The book is intended for researchers and graduate students specializing in dynamical systems who wish to have a sufficiently comprehensive view of the theory together with a working knowledge of its main techniques. The discussion of some open problems is also included in the hope that it may lead to further developments. Ideally, readers should have some familiarity with the basic notions and results of ergodic theory and hyperbolic dynamics at the level of an introductory course in the area, though the initial chapters also review all the necessary material.
Author: A.V. Bolsinov
Publisher: Springer
ISBN:
Category : Mathematics
Languages : en
Pages : 344
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Book Description
From Moscow State University, Bolsinov (computer methods) and Fomenko (differential geometry) present a new approach to the qualitative analysis of the particular type of geodesic flow of Riemannian metrics on manifolds based on the theory of topological classification of integrable Hamiltonian systems. They begin by introducing the qualitative theory of integrable Hamiltonian systems, then discuss the class of integrable geodesic flows on two-dimensional surfaces from both the classical and contemporary perspectives. They classify the flows according to equivalence relations, such as isometry, the Liouville equivalence, the smooth and continuous trajectory equivalence, and the geodesic equivalence. They also explain the new technique that makes such classification possible. Many of their results have not been published before. The Russian original is Geometriia i topologiia integriruemykh geodezicheskikh potokov na poverkhnostiakhAnnotation copyrighted by Book News, Inc., Portland, OR
