INTEGRABLE MAPPINGS AND POLYNOMIAL-GROWTH

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable ve rtex-models in lattice statistical mechanics. These involutions corres pond respectively to two kinds of transformations on q x q matrices: t he inversion of the q x q matrix and an (involutive) permutation of th e entries of the matrix. We concentrate on the case where these permut ations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each c lass we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for q = 4, actually lie on elliptic curves. This analysis also provides examp les of integrable mappings in arbitrary dimension, even infinite. More over, for two classes, the homogeneous polynomials are shown to satisf y non trivial non-linear recurrences. The relations between factorizat ions of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.