FACTORIZATION PROPERTIES OF BIRATIONAL MAPPINGS

We analyse birational mappings generated by transformations on q x q m atrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q x q ma trix, Remarkable factorization properties emerge for quite general inv olutive permutations. It is shown that factorization properties do exi st, even for birational transformations associated with noninvolutive permutations of entries of q x q matrices, and even for more general t ransformations which are rational transformations but no longer birati onal. The existence of factorization relations independent of q, the s ize of the matrices, is underlined. The relations between the polynomi al growth of the complexity of the iterations, the existence of recurs ions in a single variable and the integrability of the mappings, are s ketched for the permutations yielding these properties. All these resu lts show that permutations of the entries of the matrix yielding facto rization properties are not so rare. In contrast, the occurrence of re cursions in a single variable, or of the polynomial growth of the comp lexity are, of course, less frequent but not completely exceptional.