  We examine a family of discrete second-order systems which are integrable
through reduction to a linear system. These systems were previously identified
using the singularity confinement criterion. Here we analyse them using the
more stringent criterion of nonexponential growth of the degrees of the
iterates. We show that the linearisable mappings are characterised by a very
special degree growth. The ones linearisable by reduction to projective systems
exhibit zero growth, i.e. they behave like linear systems, while the remaining
ones (derivatives of Riccati, Gambier mapping) lead to linear growth. This
feature may well serve as a detector of integrability through linearisation.
