This textbook offers a uniquely accessible introduction to flows on compact surfaces, filling a gap in the existing literature. The book can be used for a single semester course and/or for independent study. It demonstrates that covering spaces provide a suitable and modern setting for studying the structure of flows on compact surfaces. The thoughtful treatment of flows on surfaces uses topology (especially covering spaces), the classification of compact surfaces, and Euclidean and hyperbolic rigid motions to establish structural theorems that describe flows on surfaces generally. Several of the topics from dynamical systems that appear in this book (e.g., fixed points, invariant sets, orbits, almost periodic points) also appear in the many subareas of dynamical systems. The book successfully presents the reader with a self-contained introduction to dynamical systems or an expansion of one's existing knowledge of the field. Prerequisites include completion of a graduate-level topology course; a background in dynamical systems is not assumed.
Author(s): Nelson G. Markley , Mary Vanderschoot
Series: Birkhäuser Advanced Texts
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 362
Tags: Dynamical Systems, Flows, Covering Spaces, Hyperbolic Geometry, Lifts
Preface
Contents
1 Dynamical Systems
1.1 Continuous Group Actions
1.2 Flows and Cascades
1.3 Beck's Theorem
2 Flows and Covering Spaces
2.1 Topological Manifolds
2.2 Lifting Flows
2.3 Compact Connected Surfaces
3 A Family of Examples
3.1 Suspension Flows
3.2 Cascades on the Circle
3.3 Denjoy Flows on the Torus
4 Local Sections
4.1 Bebutoff's Theorem
4.2 Whitney's Theorem
4.3 Classical Applications
5 Flows on the Torus
5.1 Weil's Theorem
5.2 Control Curves
5.3 Geometry of Recurrent Orbits
6 Hyperbolic Geometry
6.1 Poincaré Disk Model
6.2 Properties of Rigid Motions
6.3 Groups of Rigid Motions
7 Flows and Hyperbolic Geometry
7.1 Fuchsian Groups
7.2 Constructing Compact Dirichlet Regions
7.3 Lifts of Closed Curves Again
8 Lifts and Limits
8.1 The Anosov Dichotomy
8.2 Winding
8.3 Omega Limit Points at Infinity
8.4 Geometry of Almost Periodic Orbits
9 Recurrent Orbit Closures
9.1 Covering Space Criteria
9.2 Maier's Theorems
9.3 Counting Recurrent Orbit Closures
10 Existence of Transitive Flows
10.1 Constructing Rectangular Surfaces
10.2 Constructing Flows from Orbits
10.3 Transitivity
10.4 Locally Circular Cascades and Flows
Bibliography
Index of Special Symbols
Index
