Blending two discrete integrability criteria: singularity confinement and
We confront two integrability criteria for rational mappings. The first
the singularity confinement based on the requirement that every singularity,
spontaneously appearing during the iteration of a mapping, disappear
some steps. The second recently proposed is the algebraic entropy criterion
associated to the growth of the degree of the iterates. The algebraic
results confirm the previous findings of singularity confinement on
Painlev\'e equations. The degree-growth methods are also applied to
systems. The result is that systems integrable through linearisation
slower growth than systems integrable through isospectral methods.
provide a valuable detector of not just integrability but also of the
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