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Contents
Contents
Dynamical Systems
Lecturer: Sebastian van Strien
October, 2000
Contents
Aims and Objectives
Acknowledgment
Circle Maps
Introduction
Rotation Numbers and Poincaré's Theorem
An Alternative Formulation of Rotation Number
Denjoy's Theorem
Family of Circle Maps and Arnol'd Tongues
Structural Stability
Interval Maps
Sharkovskii's Theorem
Doubling Map and the Full Quadratic Map
Some Ergodic Theory
Examples in Two Dimensions
The Horseshoe Map
The Linear Anosov Map
Hyperbolic Dynamics
The Theorem of Hartman-Grobman
Stability of Hyperbolic Periodic Points
Stable and Unstable Manifolds
-Lemma
Stable Manifold Theorem for General Invariant Sets
The Shadowing Lemma
An Application of the Shadowing Lemma
Examples
The Smale Attractor (Solenoid)
Homoclinic Intersections: The Occurrence of Smale's Horseshoe
-Explosion
The Markov Partition Associated to a Linear Anosov Map
Exercises
Assignment 1
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Exam Questions
Beyond the Scope of This Course
About this document ...
Sebastian van Strien 2001-11-09
-Lemma
-Explosion