Discrete Integrable Systems
QRT Maps and Elliptic Surfaces
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Beschreibung
The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the author’s insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodaira’s theory of elliptic surfaces and the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraic- and complex analytic geometry, different categories of readers will be able to become familiar with any selected interest in the book without having to make an extensive journey through the literature.
This book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard, is examined in detail. The link between elliptic fibrations and completely integrable Hamiltonian systems is also discussed.
The book begins with a comprehensive overview of the subject, including QRT maps, singularity confinement, automorphisms of rational elliptic surfaces, action on homology classes, and periodic QRT maps. Later chapters cover these topics and more in detail.
While QRT maps will be familiar to specialists in algebraic geometry, the present volume makes the subject accessible to mathematicians and graduate students in a classroom setting or for self-study.
Makes the theory of QRT maps accessible to non-specialists in algebraic geometry
May be used as introduction to the theory of general elliptic surfaces
Applies theory to Poncelet mappings, the elliptic billiard, and difference equations from mathematical physics
Almost everything can be explicitly computed
This book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard, is examined in detail. The link between elliptic fibrations and completely integrable Hamiltonian systems is also discussed. The book begins with a comprehensive overview of the subject, including QRT maps, singularity confinement, automorphisms of rational elliptic surfaces, action on homology classes, and periodic QRT maps. Later chapters cover these topics and more in detail. While QRT maps will be familiar to specialists in algebraic geometry, the present volume makes the subject accessible to mathematicians and graduate students in a classroom setting or for self-study.
Autor*in
J.J. Duistermaat
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From the reviews:
“The book … takes us on a tour of the field and provides us with a wealth of applications of various concepts coming from algebraic and analytic geometry to dynamical systems. … the book gives an exhaustive analysis of discrete algebraically integrable maps in two dimensions. The bibliography contains over 200 references, and covers recent works as well as fundamental titles, making it an inescapable encyclopaedic reference on the subject.” (Claude M. Viallet, Bulletin of the London Mathematical Society, Vol. 45 (4), August, 2013)
“This is an excellent book for graduates and researchers who have some knowledge of integrable systems and a rudimentary understanding of algebraic geometry. … Overall, this book delivers an excellent overview of a geometric interpretation of this class of discrete integrable systems, and is written in a manner that is fairly accessible to a postgraduate student or keen researcher in integrable systems.” (C. M. Ormerod, G. R. W. Quispel and J. A. G. Roberts, SIAM Review, Vol. 54 (1), 2012)
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