Growth and integrability in discrete systems

We introduce a new discrete integrability criterion inspired from the recent findings of Ablowitz and collaborators. This criterion is based on the study of the growth of some characteristic of the solutions of a mapping, using Nevanlinna theory. Since the practical implementation of the latter does not always lead to a clear-cut answer, we complement the growth criterion by the singularity confinement property. This combination turns out to be particularly efficient. Its application allows us to recover the known forms of the discrete Painleve equations and to show that no new ones may exist within a given parametrization.