Equivariant Degree Theory

Jorge Ize, Alfonso Vignoli

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English

De Gruyter
16 April 2003

Non-linear science; Physical geography & topography

Series: De Gruyter Series in Nonlinear Analysis & Applications

This volume presents a 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 explicitly computed and the effects of symmetry breaking, products and composition are thoroughly 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.

By: Jorge Ize, Alfonso Vignoli
Imprint: De Gruyter
Country of Publication: Germany
Edition: Reprint 2012
Volume: 8
Dimensions: Height: 240mm, Width: 170mm, Spine: 25mm
Weight: 754g
ISBN: 9783110175509
ISBN 10: 3110175509
Series: De Gruyter Series in Nonlinear Analysis & Applications
Pages: 380
Publication Date: 16 April 2003
Recommended Age: College Graduate Student
Audience: Professional and scholarly , Professional and scholarly , Undergraduate , Undergraduate
Format: Hardback
Publisher's Status: Active

Reviews for Equivariant Degree Theory

Due to the included results and examples and to the self-contained and unifying approach, this book can be helpful to researchers and postgraduate students working in nonlinear analysis, differential equations, topology, and quantitative aspects of applied mathematics. Csaba Varga in: Studia Universitates Babes-Bolyai 1/2007