Integrable lattice maps: Q5 a rational version of Q4 arXiv:0802.0294

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 calculation of algebraic entropy. We first explain the setting, i.e. what are two dimensional lattice maps on a square lattice (quad maps), and describe two characteristics of integrability of such systems, respectively Lax pair and consistency [2, 7], with the important (generic) example Q4, and vanishing of algebraic entropy [8, 9, 10, 11], which, as we will show, provides a natural generalisation of Q4, baptised Q5. We explain the factorization process of the iterates at the origin of the vanishing of the entropy, and present some directions for further investigations.