A classification of two-dimensional integrable mappings and rational elliptic surfaces

Abstract

We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which exchange fibers. We also show an equivalent condition when a generalized Halphen surface becomes a Halphen surface of index m.

Studies on spaces of initial conditions for non-autonomous mappings of the plane

“…Step 6 It is clear that if f is diagonalizable, then its Jordan normal form is (3.1). Thus it is sufficient to show that if f is not diagonalizable, then its Jordan normal form is (3.2).…”

“…. , P (6) , Q (1) , Q (2) , Q (3) . After sufficient steps, however, these points again become curves.…”

A note on minimization of rational surfaces obtained from birational dynamical systems

“…Step 1 of Appendix of [3] and the authors' paper [1]). Let C be a curve in the linear system |kθ| (such C exists [2]).…”

Birational maps that send biquadratic curves to biquadratic curves

“…This communication concerns generalizations of discrete integrable systems in the form of integrable maps of the plane that have received a lot of attention in the recent literature. We consider birational maps M : (x, y) → (x , y ) of the form (1) M : x = y, y = R(x, y).…”

“…These examples fall into two categories, which we label the intrafibration case and the interfibration case. 1 In the intrafibration case, M sends a biquadratic curve labelled t in (3) to a different one in the same family, labelled τ = f (t) = t, so (5) M : B(x, y; t) = 0 ⇒ B(x , y ; τ ) = 0.…”