# Dynamical Systems on 2- and 3-Manifolds

Part of the Developments in Mathematics book series (DEVM, volume 46)

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Part of the Developments in Mathematics book series (DEVM, volume 46)

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.

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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.

diffeomorphisms on 2-manifolds diffeomorphisms on 3-manifolds qualitative theory of differential equations discrete dynamical systems Morse-Smale diffeomorphisms Morse-Lyapunov functions dynamical systems on manifolds

- DOI https://doi.org/10.1007/978-3-319-44847-3
- Copyright Information Springer International Publishing Switzerland 2016
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-44846-6
- Online ISBN 978-3-319-44847-3
- Series Print ISSN 1389-2177
- Series Online ISSN 2197-795X
- Buy this book on publisher's site