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Degree Theory for Equivariant Maps, the General \(S^1\)-Action
Jorge Ize, Ivar Massabo, and Alfonso Vignoli
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Memoirs of the American Mathematical Society
1992; 179 pp; softcover
Volume: 100
ISBN-10: 0-8218-2542-9
ISBN-13: 978-0-8218-2542-6
List Price: US$36
Individual Members: US$21.60
Institutional Members: US$28.80
Order Code: MEMO/100/481
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This work is devoted to a detailed study of the equivariant degree and its applications for the case of an \(S^1\)-action. This degree is an element of the equivariant homotopy group of spheres, which are computed in a step-by-step extension process. Applications include the index of an isolated orbit, branching and Hopf bifurcation, and period doubling and symmetry breaking for systems of autonomous differential equations. The authors have paid special attention to making the text as self-contained as possible, so that the only background required is some familiarity with the basic ideas of homotopy theory and of Floquet theory in differential equations. Illustrating in a natural way the interplay between topology and analysis, this book will be of interest to researchers and graduate students.

Readership

Researchers and graduate students who wish to learn about the interplay between topology and analysis.

Table of Contents

  • Preliminaries
  • Extensions of \(S^1\)-maps
  • Homotopy groups of \(S^1\)-maps
  • Degree of \(S^1\)-maps
  • \(S^1\)-index of an isolated non-stationary orbit and applications
  • Index of an isolated orbit of stationary solutions and applications
  • Virtual periods and orbit index
  • Appendix: Additivity up to one suspension
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