ELLIPTIC-CURVES FROM FINITE-ORDER RECURSIONS OR NONINVOLUTIVE PERMUTATIONS FOR DISCRETE DYNAMICAL-SYSTEMS AND LATTICE STATISTICAL-MECHANICS

We study birational mappings generated by matrix inversion and permuta tions of the entries of q x q matrices. For q = 3 we have performed a systematic examination of all the birational mappings associated with permutations of 3 x 3 matrices in order to find integrable mappings an d some finite order recursions. This exhaustive analysis gives, among 30 462 classes of mappings, 20 classes of integrable birational mappin gs, 8 classes associated with integrable recursions and 44 classes yie lding finite order recursions. An exhaustive analysis (with a constrai nt on the diagonal entries) has also been performed for 4 x 4 matrices : we have found 880 new classes of mappings associated with integrable recursions. We have visualized the orbits of the birational mappings corresponding to these 880 classes. Most correspond to elliptic curves and very few to surfaces or higher dimensional algebraic varieties. A ll these new examples show that integrability can actually correspond to non-involutive permutations. The analysis of the integrable cases s pecific of a particular size of the matrix and a careful examination o f the non-involutive permutations, shed some light on the integrabilit y of such birational mappings.