We consider a family of birational transformations of two variables, depending on one parameter, for which simple rational expressions with integer coefficients, for the exact expression of the dynamical zeta function, have been conjectured. Moreover, an equality between the (asymptotic of the)...

We propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer coefficients. This yields an algebraic value...

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters (ε and α), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping...

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive...

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six...

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q × q matrices: the inversion of...