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Dynamical Systems on 2- and 3-Manifolds

Part of the Developments in Mathematics Book Series (volume 46)

DOI: 10.1007/978-3-319-44847-3

ISBN: 978-3-319-44846-6

Publisher: Springer International Publishing - Springer Journals

Nov 12, 2016

1 - 10 of 10 Chapters

LitStream

Chapter 1Introduction to Dynamical Systems

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

Pages: 1 –26

[In this chapter we present without proofs notions and facts on the dynamical systems which are necessary to understand this book. We recall the notion of an invariant set and show the most important examples of such sets: fixed and periodic points, \documentclass[12pt]{minimal}...
LitStream

Chapter 2General Properties of the Morse–Smale Diffeomorphisms

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

Pages: 27 –55

[In this chapter we consider an important class of discrete structurally stable systems which adequately describe processes with regular dynamics, Morse-Smale diffeomorphisms. We present with proofs the properties of Morse-Smale diffeomorphisms which are necessary for the topological...
LitStream

Chapter 3The Topological Classification of the Gradient-Like Diffeomorphism on Surfaces

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

Pages: 57 –75

[In this chapter we study the periodic data (the period, the Morse index, the orientation type) of the periodic orbits of gradient-like diffeomorphisms on orientable surfaces. Though such diffeomorphisms are similar in many ways to structurally stable flows on surfaces they have one property...
LitStream

Chapter 4Wild Embedding of the Separatrices into 3-Manifolds and Pixton Diffeomorphism

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

Pages: 77 –107

[The Morse-Smale diffeomorphisms on 3-manifolds are in marked contrast to the Morse-Smale flows and diffeomorphisms on 2-manifolds because of the possibility of wild embedding of the separatrices of the saddle points. In this chapter we prove the criteria of the tame embedding of 1- and...
LitStream

Chapter 5The Classification of the Gradient-Like Diffeomorphisms on 3-Manifolds

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

Pages: 109 –118

[In the classical papers by S. Smale and J. Palis the proof of structural stability of Morse-Smale diffeomorphisms was based on the construction of a system of tubular neighborhoods. We present the similar construction for gradient-like 3-diffeomorphisms using the idea of representation of the...
LitStream

Chapter 6Interrelation Between the Dynamics of Morse–Smale Diffeomorphisms and the Topology of the Ambient 3-Manifold

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

Pages: 119 –130

[In this chapter we state some interrelations between the topology of the ambient manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 7An Energy Function for Morse–Smale Diffeomorphisms on 3-Manifolds

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

Pages: 131 –147

[Following the ideas of A. Lyapunov C. Conley introduced the notion of a Lyapunov function for a dynamical system (see Definition 7.1). In 1978 he proved the existence of a continuous Lyapunov function for every dynamical system [2]. This result is called the fundamental theorem of dynamical...
LitStream

Chapter 8The Properties of Nontrivial Basic Sets of A-Diffeomorphisms Related to Type and Dimension

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

Pages: 149 –165

[In this chapter we study orientation preserving A-diffeomorphisms on an orientable compact manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
LitStream

Chapter 9The Classification of Nontrivial Basic Sets of A-Diffeomorphisms of Surfaces

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

Pages: 167 –216

[The first problem which arises when studying a topological classification of A-homeomorphisms with nontrivial basic sets is the problem to find a suitable topological invariant which can adequately reflect the restriction of the diffeomorphism to the basic set as well as the embedding of the...
LitStream

Chapter 10Basic Topological Concepts of Dynamical Systems

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

Pages: 217 –286

[The theory of dynamical systems extensively uses a lot of concepts and tools from other branches of mathematics: topology, algebra, geometry etc. In this chapter we review the basic definitions and facts, necessary for understanding the presented results. We begin the fundamental notions of a...

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