Equivariant Degree Theory
Equivariant Degree Theory
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DOI：
10.1515/9783110200027
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发表时间：
2003-04
期刊：
影响因子：
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通讯作者：
J. Ize;A. Vignoli
中科院分区：
文献类型：
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作者：
J. Ize;A. Vignoli
本书提出了一种新的与一组对称性交换的映射的度理论。该度数不再是单个整数，而是两个球体之间映射的等变同伦类群的元素，并且取决于空间的轨道类型。作者从基本事实开始，通过独立的演示完整地阐述了该学位的理论和应用。第一章讲解了文中所需的表示论、同伦论和微分方程的基本工具。然后定义度并推导其主要抽象属性。下一部分致力于研究球面的等变同伦群以及交换行为情况下等变映射的分类。这些群经过明确的计算，对称性破缺、产物和成分的影响也得到了彻底的研究。最后一部分涉及孤立轨道和驻点孤立环的等变指数的计算。这里考虑了各种情况下的微分方程：对称破缺、强迫、倍周期、扭曲轨道、一阶积分、梯度等。研究了哈密顿系统（特别是弹簧摆系统）的周期解以及所有这些情况下的 Hopf 分岔。
This book presents a new degree theory for maps which commute with a group of symmetries. This degree is no longer a single integer but an element of the group of equivariant homotopy classes of maps between two spheres and depends on the orbit types of the spaces. The authors develop completely the theory and applications of this degree in a self-contained presentation starting with only elementary facts. The first chapter explains the basic tools of representation theory, homotopy theory and differential equations needed in the text. Then the degree is defined and its main abstract properties are derived. The next part is devoted to the study of equivariant homotopy groups of spheres and to the classification of equivariant maps in the case of abelian actions. These groups are explicitely computed and the effects of symmetry breaking, products and composition are thorougly studied. The last part deals with computations of the equivariant index of an isolated orbit and of an isolated loop of stationary points. Here differential equations in a variety of situations are considered: symmetry breaking, forcing, period doubling, twisted orbits, first integrals, gradients etc. Periodic solutions of Hamiltonian systems, in particular spring-pendulum systems, are studied as well as Hopf bifurcation for all these situations.