FROM INTEGRABILITY TO WEAK CHAOS

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. The iterations corresponding to the three other classes are include d in higher dimensional non-trivial algebraic varieties. For many init ial conditions in the parameter space these orbits lie on (transcenden tal) curves, and finally explode in these higher dimensional varieties . These transformations act on fifteen (or q2 - 1) variables, however one can associate to them remarkably simple non-linear recurrences bea ring on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings.