The road to the discrete analogue of the Painlevé property: Nevanlinna meets singularity confinement

The question of integrability of discrete systems is analyzed in the light of the recent findings of Ablowitz et al., who have conjectured that a fast growth of the solutions of a difference equation is an indication of nonintegrability. The study of the behaviour of the solutions of a mapping is based on the theory of Nevanlinna. In this paper, we show how this approach can be implemented in the case of second-order mappings which include the discrete Painlevé equations. Since the Nevanlinna approach does offer only a necessary condition which is not restrictive enough, we complement it by the singularity confinement requirement, first in an autonomous setting and then for deautonomisation. We believe that this three-tiered approach is the closest one can get to a discrete analogue of the Painlevé property.