For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations… (More)

We introduce a " pre-Bethe-Ansatz " system of equations for three dimensional vertex models. We bring to the light various algebraic curves of high genus and discuss some situations where these… (More)

We describe deformations of non-linear (birational) representations of discrete groups generated by involutions, having their origin in the theory of the symmetric five-state Potts model. One of the… (More)

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group A (1) 2. The existence of this symmetry explains the presence of a spectral… (More)

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach… (More)

We give a rational form of a generic two-dimensional “quad” map, containing the so-called Q4 case [1, 2, 3, 4, 5, 6], but whose coefficients are free. Its integrability is proved using the… (More)

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the… (More)

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective… (More)

We give a rational form of a generic two-dimensional “quad” map, containing the so-called Q4 case [1, 2, 3, 4, 5, 6], but whose coefficients are free. Its integrability is proved using the… (More)