ANZIAM  J.  44 (2002), 21-32
What is the discrete analogue of the Painlevé property?

A. Ramani
  Ecole Polytechnique
  UMR 7644
  91128 Palaiseau
B. Grammaticos
  Université Paris VII
  Tour 24-14
  5 étage, case 7021
  75251 Paris

We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.
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