SINGULARITY, COMPLEXITY, AND QUASI-INTEGRABILITY OF RATIONAL MAPPINGS

We investigate global properties of the mappings entering the descript ion of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We gi ve an algorithmic analysis of the structure of invariants of such mapp ings. We discuss some characteristic conditions for their (quasi)-inte grability, and in particular its links with their singularities (in th e 2-plane). Finally, we describe some of their properties qua dynamica l systems, making contact with Arnol'd's notion of complexity, and exe mplify remarkable behaviours.