The QRT Map -- The Pencil of Biquadratic Curves in -- The QRT surface -- Cubic Curves in the Projective Plane -- The Action of the QRT Map on Homology -- Elliptic Surfaces -- Automorphisms of Elliptic Surfaces -- Elliptic Fibrations with a Real Structure -- Rational elliptic surfaces -- Symmetric QRT Maps -- Examples from the Literature -- Appendices.

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.