Mathematics-Dynamical Systems

Harmonic maps and integrable systems

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Title (user) : Harmonic maps and integrable systemsISBN : 3528065540,9783528065546DDC : 514/.74LCC : QA614.73 .H36 1994GoogleBook ID : tVTvAAAAMAAJOpenLibrary ID : OL1176726MSeries : Vieweg Advanced Studies in Computer ScienceAuthors (user) : Allan P. FordyAuthors (google) : Allan P. Fordy,John C. WoodPublisher : Friedr Vieweg & Sohn VerlagsgesellschaftLanguage : EnglishPublication Date : 1994Scanned : yes (300 DPI)File Format : djvuCategories : Mathematics------------------------------------------Description (user) : Harmonic maps are maps between Riemannian or pseudo-Riemannian manifolds which extremize a natural energy integral. They have found many applications, for example, to the theory of minimal and constant mean curvature suface. In physics they arise as the non-linear sigma and chiral models of particle physics. Recently, there has been an explosion of interest in applying the methods to ingrable systems to find and study harmonic maps. Bringing together experts in the field of harmonic maps and integrable systems to give a coherent account of this subject, this book starts with introductory articles, so that the book is self-contained. It should be of interest to graduate students and researchers interested in applying integrable systems to variational problems, and could form the basis of a graduate course.------------------------------------------Description (google) : We obtain stability and structural results for equivariant diffeomorphisms which are hyperbolic transverse to a comp...
Description

Title (user) : Harmonic maps and integrable systems

ISBN : 3528065540,9783528065546

DDC : 514/.74

LCC : QA614.73 .H36 1994

GoogleBook ID : tVTvAAAAMAAJ

OpenLibrary ID : OL1176726M

Series : Vieweg Advanced Studies in Computer Science

Authors (user) : Allan P. Fordy

Authors (google) : Allan P. Fordy,John C. Wood

Publisher : Friedr Vieweg & Sohn Verlagsgesellschaft

Language : English

Publication Date : 1994

Scanned : yes (300 DPI)

File Format : djvu

Categories : Mathematics


------------------------------------------

Description (user) :
Harmonic maps are maps between Riemannian or pseudo-Riemannian manifolds which extremize a natural energy integral. They have found many applications, for example, to the theory of minimal and constant mean curvature suface. In physics they arise as the non-linear sigma and chiral models of particle physics. Recently, there has been an explosion of interest in applying the methods to ingrable systems to find and study harmonic maps. Bringing together experts in the field of harmonic maps and integrable systems to give a coherent account of this subject, this book starts with introductory articles, so that the book is self-contained. It should be of interest to graduate students and researchers interested in applying integrable systems to variational problems, and could form the basis of a graduate course.


------------------------------------------

Description (google) :
We obtain stability and structural results for equivariant diffeomorphisms which are hyperbolic transverse to a compact (connected or finite) Lie group action and construct `$Gamma$-regular' Markov partitions which give symbolic dynamics on the orbit space. We apply these results to the situation where $Gamma$ is a compact connected Lie group acting smoothly on $M$ and $F$ is a smooth (at least $C^2$) $Gamma$-equivariant diffeomorphism of $M$ such that the restriction of $F$ to the $Gamma$- and $F$-invariant set $Lambdasubset M$ is partially hyperbolic with center foliation given by $Gamma$-orbits. On the assumption that the $Gamma$-orbits all have dimension equal to that of $Gamma$, we show that there is a naturally defined $F$- and $Gamma$-invariant measure $u$ of maximal entropy on $Lambda$ (it is not assumed that the action of $Gamma$ is free). In this setting we prove a version of the Livsic regularity theorem and extend results of Brin on the structure of the ergodic components of compact group extensions of Anosov diffeomorphisms. We show as our main result that generically $(F,Lambda,u)$ is stably ergodic (openness in the $C^2$-topology). In the case when $Lambda$ is an attractor, we show that $Lambda$ is generically a stably SRB attractor within the class of $Gamma$-equivariant diffeomorphisms of $M$.

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