44 (2002), 21-32|
What is the discrete analogue of the Painlevé property?
Université Paris VII
5 étage, case 7021
| 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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