Blending two discrete integrability criteria: singularity confinement and algebraic entropy


Stéphane Lafortune

We confront two integrability criteria for rational mappings. The first is
the singularity confinement based on the requirement that every singularity,
spontaneously appearing during the iteration of a mapping, disappear after
some steps. The second recently proposed is the algebraic entropy criterion
associated to the growth of the degree of the iterates. The algebraic entropy
results confirm the previous findings of singularity confinement on discrete
Painlev\'e equations. The degree-growth methods are also applied to linearisable
systems. The result is that systems integrable through linearisation have a
slower growth than systems integrable through isospectral methods. This may
provide a valuable detector of not just integrability but also of the precise integration
method. We propose an extension of the Gambier mapping in $N$ dimensions.
Finally a dual strategy for the investigation of the integrability of discrete systems
is proposed based on both singularity confinement and the low growth requirement.