# Degree Theory for Equivariant Maps, the General \(S^1\)-Action

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*Jorge A Ize; Alfonso Vignoli; Ivar Massabo*

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.

#### Table of Contents

# Table of Contents

## Degree Theory for Equivariant Maps, the General $S^{1}$-Action

- TABLE OF CONTENTS v6 free
- INTRODUCTION 112 free
- CHAPTER ONE: PRELIMINARIES 1324
- CHAPTER TWO: EXTENSIONS OF 5[sup(1)]-MAPS 2132
- CHAPTER THREE: HOMOTOPY GROUPS OF S[sup(1)]-MAPS 4152
- CHAPTER FOUR: DEGREE OF S[sup(1)]-MAPS 6980
- CHAPTER FIVE: S[sup(1)]-INDEX OF AN ISOLATED NON-STATIONARY ORBIT AND APPLICATIONS 101112
- 5.1. The case p ≥ 1 102113
- 5.2. The case p = 0 103114
- 5.3. p = 0, hyperbolic orbits 106117
- 5.4. Autonomous differential equations 114125
- 5.5. Gradient maps 116127
- 5.6. Differential equations with fixed period 116127
- 5.7. Differential equations with first integrals 118129
- 5.8 Symmetry breaking for differential equations 120131

- CHAPTER SIX: INDEX OF AN ISOLATED ORBIT OF STATIONARY SOLUTIONS AND APPLICATIONS 125136
- CHAPTER SEVEN: VIRTUAL PERIODS AND ORBIT INDEX 159170
- APPENDIX: ADDITIVITY UP TO ONE SUSPENSION 173184
- REFERENCES 177188