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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.
List of contents
1. Dynamical Systems.- 2. Flows and Covering Spaces.- 3. A Family of Examples.- 4. Local Sections.- 5. Flows on the Torus.- 6. Hyperbolic Geometry.- 7. Flows and Hyperbolic Geometry.- 8. Lifts and Limits.- 9. Recurrent Orbit Closures.- 10. Existence of Transitive Flows.
About the author
Nelson G. Markley was a professor of mathematics at the University of Maryland for more than twenty-five years and also served as provost and senior vice president at Lehigh University. He authored numerous journal articles in the area of dynamical systems as well as textbooks on differential equations, topological groups, and probability. He received his PhD from Yale University.
¿Mary Vanderschoot is a professor of mathematics at Wheaton College (IL). She holds a PhD in topological dynamical systems from the University of Maryland. Nelson Markley was her PhD advisor.
Report
"The authors make this monograph as accessible as possible. It is written to be usable as a text which is self-contained ... . This book caps decades of research on the subject and was written ... that it might foster further investigation. ... The book pulls together a field which through war, cold war and happenstance had been disrupted and disconnected ... . In doing so, this monograph coherently weaves together strands of a loose 90-year old tangle of ideas." (Boris Hasselblatt, Mathematical Reviews, April, 2024)
