From integrability to weak chaos

Abstract

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. The iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. For many initial conditions in the parameter space these orbits lie on (transcendental) curves, and finally explode in these higher dimensional varieties. These transformations act on fifteen (or q 2−1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings.