Algebraic-geometry approach to integrability of birational plane mappings. Integrable birational quadratic reversible mappings. I

Abstract

Using classic results of algebraic geometry for birational plane mappings in plane CP2we present a general approach to algebraic integrability of autonomous dynamical systems in C2with discrete time and systems of two autonomous functional equations for meromorphic functions in one complex variable defined by birational maps in C2. General theorems defining the invariant curves, the dynamics of a birational mapping and a general theorem about necessary and sufficient conditions for integrability of birational plane mappings are proved on the basis of a new idea — a decomposition of the orbit set of indeterminacy points of direct maps relative to the action of the inverse mappings. A general method of generating integrable mappings and their rational integrals (invariants)Iis proposed. Numerical characteristicsNkof intersections of the orbitsΦn−kOiof fundamental or indeterminacy pointsOiεO∩S, of mappingΦn, whereO= {Oi} is the set of indeterminacy points ofΦnandSis a similar set for invariantI, with the corresponding setO′ ∩S, whereO′ = {O′i} is the set of indeterminacy points of inverse mappingΦn−1, are introduced. Using the method proposed we obtain all nine integrable multiparameter quadratic birational reversible mappings with the zero fixed point and linear projective symmetryS = C Λ C−1, Λ = diag(±1), with rational invariants generated by invariant straight lines and conics. The relations of numbersNkwith such numerical characteristics of discrete dynamical systems as the Arnold complexity and their integrability are established for the integrable mappings obtained. The Arnold complexities of integrable mappings obtained are determined. The main results are presented in Theorems 2–5, in Tables 1 and 2, and in Appendix A.