Factorization properties of birational mappings

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations. It is shown that factorization properties do exist, even for birational transformations associated with noninvolutive permutations of entries of q × q matrices, and even for more general transformation which are rational transformations but no longer birational. The existence of factorization relations independent of q, the size of the matrices, is underlined. The relations between the polynomial growth of the complexity of the iterations, the existence of recursions in a single variable and the integrability of the mappings, are sketched for the permutations yielding these properties. All these results show that permutations of the entries of the matrix yielding factorization properties are not so rare. In contrast, the occurrence of recursions in a single variable, or of the polynomial growth of the complexity are, of course, less frequent but not completely exceptional.