**Graduate Studies in Mathematics**

Volume: 85;
2007;
221 pp;
Hardcover

MSC: Primary 37; 54;
Secondary 34

**Print ISBN: 978-0-8218-4234-8
Product Code: GSM/85**

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

**Electronic ISBN: 978-1-4704-2115-1
Product Code: GSM/85.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

#### Supplemental Materials

# Recurrence and Topology

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*John M. Alongi; Gail S. Nelson*

Since at least the time of Poisson,
mathematicians have pondered the notion of recurrence for differential
equations. Solutions that exhibit recurrent behavior provide insight
into the behavior of general solutions. In Recurrence and
Topology, Alongi and Nelson provide a modern understanding of the
subject, using the language and tools of dynamical systems and
topology.

Recurrence and Topology develops increasingly more general
topological modes of recurrence for dynamical systems beginning with
fixed points and concluding with chain recurrent points. For each
type of recurrence the text provides detailed examples arising from
explicit systems of differential equations; it establishes the general
topological properties of the set of recurrent points; and it
investigates the possibility of partitioning the set of recurrent
points into subsets which are dynamically irreducible. The text
includes a discussion of real-valued functions that reflect the
structure of the sets of recurrent points and concludes with a
thorough treatment of the Fundamental Theorem of Dynamical
Systems.

Recurrence and Topology is appropriate for mathematics graduate
students, though a well-prepared undergraduate might read most of the
text with great benefit.

#### Readership

Undergraduate and graduate students interested in dynamical systems and topology.

#### Reviews & Endorsements

The book is clear, and generally well-written, and could profitably be read by either graduate students or advanced undergraduates. The presentation is extremely detailed.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Recurrence and Topology

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Foreword vii8 free
- Chapter 1. Flows 112 free
- Chapter 2. Recurrent Points 2334
- Chapter 3. Irreducible Sets 99110
- Chapter 4. Test Functions 139150
- §4.1. Potential Functions on Euclidean Spaces and Surfaces 139150
- §4.2. Hamiltonian Functions 148159
- §4.3. Invariant Functions 161172
- §4.4. Weak and Strong Lyapunov Functions 162173
- §4.5. Potential Functions on Riemannian Manifolds 167178
- §4.6. Morse Functions 172183
- §4.7. Complete Lyapunov Functions 176187
- §4.8. The Fundamental Theorem of Dynamical Systems 178189
- §4.9. Exercises 191202

- Afterword 197208
- Appendix A. Discrete Dynamical Systems 201212
- Appendix B. Circle Rotations 205216
- Appendix C. The Hausdorff Metric 209220
- Bibliography 213224
- Index 217228
- Back Cover Back Cover1233