Ebook: Discrete integrable systems: QRT maps and elliptic surfaces

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.

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The study of dynamical systems is a well established field. This book provides a panorama of several aspects of interest to mathematicians and physicists. It collects the material of several courses at the graduate level given by the authors, avoiding detailed proofs in exchange for numerous illustrations and examples. Apart from common subjects in this field, a lot of attention is given to questions of physical measurement and stochastic properties of chaotic dynamical systems Introduction.- The QRT Map.- Complex Projective Curves.- The QRT Surface.- Pencils of Cubic Curves in the Projective Plane.- The Action of the QRT Map on Homology Classes.- Elliptic Surfaces.- Rational Elliptic Surfaces.- Examples from the Literature.- Appendices.- Index