Baxterization, dynamical systems, and the symmetries of integrability

Abstract

We resolve the 'baxterization' problem with the help of the au-tomorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, ...) equations. This infinite group of symmetries is realized as a non-linear (bira-tional) Coxeter group acting on matrices, and exists as such, beyond the narrow context of strict integrability. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the growth of the complexity of iterations.