Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

Abstract: For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.

Abstract: Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.

Abstract: Technology of data collection and information transmission is based on various mathematical models of encoding. The words "Geometry of information" refer to such models, whereas the words "Moufang patterns" refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.

Abstract: We constructed involutions for a Halphen pencil of index 2, and proved that the birational mapping corresponding to the autonomous reduction of the elliptic Painleve equation for the same pencil can be obtained as the composition of two such involutions.

Abstract: We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize.

Abstract: We present a geometric approach to the theory of Painleve equations based on rational surfaces Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type We classify all such surfaces X To each X, there corresponds a root subsystem of E (1) 8 inside the Picard lattice of X We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces We show that the translation part of the affine Weyl group gives rise to discrete Painleve equations, and that the above action constitutes their group of symmetries by Backlund transformations The six Painleve differential equations appear as degenerate cases of this construction In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure

Abstract: Some simple solutions of discrete soliton equations are shown to satisfy 2D mappings. We show that these belong to a recently introduced 18-parameter family of integrable reversible mappings of the plane, thus lending weight to a previous conjecture. We also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics. Finally we discuss the notion of (generalized) reversibility.

Abstract: 10.1007/978-1-4419-9126-3 Copyright owner: Springer Science+Buisness Media, LLC, 2010 Data set: Springer Source Springer Monographs in Mathematics 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 authorâ€TMs 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 Kodairaâ€TMs 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 algebraicand complex analytic geometry, different categories of readers... more Identifiers series ISSN : 1439-7382 ISBN 978-1-4419-7116-6 e-ISBN 978-1-4419-9126-3 DOI Authors Additional information Publisher Springer New York book Read online Download Add to read later Add to collection Add to followed Share Export to bibliography J.J. Duistermaat Utrecht University, Department of Mathematics, Utrecht, Netherlands Terms of service Accessibility options Report an error / abuse © 2015 Interdisciplinary Centre for Mathematical and Computational Modelling Discrete integrable systems independent: it neither relies on nor used in the proof of integrability. Section 6 is not used in the proof of integrability. It discusses more specic discrete cluster integrable systems, assuming Theorem 3.7. Proof of part i) Take a pair of matchings (M1, M2) on Γ. Let us assign to them another pair of matchings (M1, M2) on Γ. Observe that [M1] − [M2] is a 1-cycle.

Abstract: 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.