Dynamical Systems on 2- and 3-Manifolds

1. Front Matter

   Pages i-xxvi

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2. Introduction to Dynamical Systems

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 1-26

3. General Properties of the Morse–Smale Diffeomorphisms

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 27-55

4. The Topological Classification of the Gradient-Like Diffeomorphism on Surfaces

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 57-75

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

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 77-107

6. The Classification of the Gradient-Like Diffeomorphisms on 3-Manifolds

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 109-118

7. Interrelation Between the Dynamics of Morse–Smale Diffeomorphisms and the Topology of the Ambient 3-Manifold

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 119-130

8. An Energy Function for Morse–Smale Diffeomorphisms on 3-Manifolds

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 131-147

9. The Properties of Nontrivial Basic Sets of A-Diffeomorphisms Related to Type and Dimension

   • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

   Pages 149-165

10. The Classification of Nontrivial Basic Sets of A-Diffeomorphisms of Surfaces

    • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

    Pages 167-216

11. Basic Topological Concepts of Dynamical Systems

    • Viacheslav Z. Grines, Timur V. Medvedev, Olga V. Pochinka

    Pages 217-286

12. Back Matter

    Pages 287-295

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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 thebasic 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
• ordinary differential equations

“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)

• Dept. Fundamental Mathematics, Natl Research Univ. HS of Economics Dept. Fundamental Mathematics, Nizhny Novgorod, Russia

  Viacheslav Z. Grines

• Dept. Differential Equations, Mathematical and Numerical Analysis, Nizhny Novgorod State University , Nizhny Novgorod, Russia

  Timur V. Medvedev

• Dept.of Fundamental Mathematics, Natl Research Univ. HS of Economics Dept. Fundamental Mathematics, Nizhny Novgorod, Russia

  Olga V. Pochinka

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