More integrable birational mappings

We study birational mappings generated by matrix inversion and permutation of the entries of q × q matrices. For q = 3 we have performed a systematic examination of all the permutations of 3 × 3 matrices in order to find integrable mappings (of three different kinds) and finite order mappings. This exhaustive analysis gives, among 30 462 classes of mappings, 27 (new) integrable classes of birational mappings and 36 classes yielding finite order recursions associated with these mappings. An exhaustive analysis (with a constraint on the diagonal entries) has also been performed for 4 × 4 matrices: we have found 8306 new classes of integrable mappings. All these new examples show that integrability can actually correspond to non-involutive permutations. The analysis of the integrable cases specific of a particular size of the matrix and a careful examination of the non-involutive permutations, could shed some light on integrability of such birational mappings. It seems that one has the following result: the non-involutive examples are specific of a given matrix size (3 × 3 matrix …) and the permutations which yield integrable mappings for arbitrary matrix size are always involutions.