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Developments in Mathematics

Dynamical Systems on 2- and 3-Manifolds

Authors: Grines, Viacheslav Z., Medvedev, Timur V., Pochinka, Olga V.

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  • Discusses classical methods and approaches considered to be promising for the further research
  • Presents main results on the topological classification of discrete dynamical systems
  • Accessible to ambitious undergraduates, graduates, and researchers in dynamical systems and low dimensional topology
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eBook $109.00
price for USA in USD
  • ISBN 978-3-319-44847-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
Softcover $139.99
price for USA in USD
About this book

This book provides an introduction to the topological classification of smooth structurally stable diffeomorphisms on closed orientable 2- and  3-manifolds.The topological classification is one of the main problems of the theory of dynamical systems and the results presented in this book are mostly for dynamical systems satisfying Smale's Axiom A. The main results on the topological classification of discrete dynamical systems are widely scattered among many papers and surveys.  This book presents these results fluidly, systematically, and for the first time in one publication. Additionally, this book discusses the recent results on the topological classification of Axiom A diffeomorphisms focusing on the nontrivial effects of the dynamical systems on 2- and 3-manifolds. The classical methods and approaches which are considered to be promising for the further research are also discussed.<

The reader needs to be familiar with the basic concepts of the qualitative theory of dynamical systems which are presented in Part 1 for convenience. The book is accessible to ambitious undergraduates, graduates, and researchers in dynamical systems and low dimensional topology. This volume consists of 10 chapters; each chapter contains its own set of references and a section on further reading. Proofs are presented with the exact statements of the results. In Chapter 10 the authors briefly state the necessary definitions and results from algebra, geometry and topology. When stating ancillary results at the beginning of each part, the authors refer to other sources which are readily available.

Reviews

“This book addresses the topological classification of smooth structurally stable discrete dynamical systems induced by a diffeomorphism on closed and orientable two- and three-dimensional manifolds. … The book is directed at experts, those trying to track down details of prior work, and others who wish to expand their expertise. The references are extensive. The authors have attempted to point the reader to more readily available sources when the original work is more difficult to find.” (William J. Satzer, MAA Reviews, maa.org, February, 2017)


Table of contents (10 chapters)

Table of contents (10 chapters)
  • Introduction to Dynamical Systems

    Pages 1-26

    Grines, Viacheslav Z. (et al.)

  • General Properties of the Morse–Smale Diffeomorphisms

    Pages 27-55

    Grines, Viacheslav Z. (et al.)

  • The Topological Classification of the Gradient-Like Diffeomorphism on Surfaces

    Pages 57-75

    Grines, Viacheslav Z. (et al.)

  • Wild Embedding of the Separatrices into 3-Manifolds and Pixton Diffeomorphism

    Pages 77-107

    Grines, Viacheslav Z. (et al.)

  • The Classification of the Gradient-Like Diffeomorphisms on 3-Manifolds

    Pages 109-118

    Grines, Viacheslav Z. (et al.)

Buy this book

eBook $109.00
price for USA in USD
  • ISBN 978-3-319-44847-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
Softcover $139.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Dynamical Systems on 2- and 3-Manifolds
Authors
Series Title
Developments in Mathematics
Series Volume
46
Copyright
2016
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-44847-3
DOI
10.1007/978-3-319-44847-3
Hardcover ISBN
978-3-319-44846-6
Softcover ISBN
978-3-319-83149-7
Series ISSN
1389-2177
Edition Number
1
Number of Pages
XXVI, 295
Number of Illustrations
95 b/w illustrations
Topics