ALMOST INTEGRABLE MAPPINGS

We analyze birational transformations obtained from very simple algebr aic calculations, namely taking the inverse of q x q matrices and perm uting some of the entries of these matrices. We concentrate on 4 x 4 m atrices and elementary transpositions of two entries. This analysis br ings out six classes of birational transformations. Three classes corr espond to integrable mappings, their iteration yielding elliptic curve s. Generically, the iterations corresponding to the three other classe s are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental ) curves. These transformations act on fifteen (or q2 - 1) variables, however one can associate to them remarkably simple non-linear recurre nces bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-in tegrable mappings, which could provide interesting tools to analyze we ak chaos.