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Geometric Theory of Dynamical Systems: An Introduction
By J. Jr. Palis, W. de Melo


Publisher: Springer
Number Of Pages: 198
Publication Date: 1982-08-18
ISBN-10 / ASIN: 0387906681
ISBN-13 / EAN: 9780387906683
This book grew from courses and seminars taught at IMPA and several other
institutions both in Brazil and abroad, a first text being prepared for the
Xth Brazilian Mathematical Colloquium. With several additions, it later
became a book in the Brazilian mathematical collection Projeto Euclides,
published in Portuguese. A number of improvements were again made for
the present translation.
We are most grateful to many colleagues and students who provided us
with useful suggestions and, above all, encouragement for us to present these
introductory ideas on Geometric Dynamics. We are particularly thankful
to Paulo Sad and, especially to Alcides Lins Neto, for writing part of a first
set of notes, and to Anthony Manning for the translation into English.


Contents
List of Symbols
Chapter 1 Differentiable Manifolds and Vector Fields 1
1 Calculus in 6d\" and Differentiable Manifolds 1
2 Vector Fields on Manifolds 10
3 The Topology of the Space of C\'^\\infty Maps 19
4 Transversality 23
5 Structural Stability 26
Chapter 2 Local Stability 39
1 The Tubular Flow Theorem 39
2 Linear Vector Fields 41
3 Singularities and Hyperbolic Fixed Points 54
4 Local Stability 59
5 Local Classification 68
6 Invariant Manifolds - 73
7 The A-lemma (inclination Lemma). Geometrical Proof of Local Stability 80
Chapter 3 The Kupka-Smale Theorem 91
1 The Poincare Map 92
2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic 99
3 Transversality of the Invariant Manifolds 106
Chapter 4 Genericity and Stability of Morse-Smale Vector Fields 115
1 Morse-Smale Vector Fields; Structural Stability 116
2 Density of Morse-Smale Vector Fields on Orientable Surfaces 132
3 Generalizations 150
4 General Comments on Structural Stability. Other Topics 153
Appendix: Rotation Number and Cherry Flows 181
References 189
Index 195