MORE INTEGRABLE BIRATIONAL MAPPINGS

We study birational mappings generated by matrix inversion and permuta tion of the entries of qxq matrices. For q=3 we have performed a syste matic examination of all the permutations of 3x3 matrices in order to find integrable mappings (of three different kinds:) and finite order mappings. This exhaustive analysis gives, among 30462 classes of mappi ngs, 27 (new) integrable classes of birational mappings and 36 classes yielding finite order recursions associated with these mappings. An e xhaustive analysis (with a constraint on the diagonal entries) has als o been performed for 4x4 matrices: we have found 8306 new classes of i ntegrable mappings. All these new examples show that integrability can actually correspond to non-involutive permutations. The analysis of t he integrable cases specific of a particular size of the matrix and a careful examination of the non-involutive permutations, could shed som e light on integrability of such birational mappings. It seems that on e has the following result: the non-involutive examples are specific o f a given matrix size (3x3 matrix...) and the permutations which yield integrable mappings for arbitrary matrix size are always involutions.