  A set of matrices is said to have the finiteness property if the maximal rate
of exponential growth of long products of matrices drawn from that set is
realised by a periodic product. The extent to which the finiteness property is
prevalent among finite sets of matrices is the subject of ongoing research. In
this article we give a condition on a finite irreducible set of matrices which
guarantees that the finiteness property holds not only for that set, but also
for all sufficiently nearby sets of equal cardinality. We also prove a theorem
giving conditions under which the Barabanov norm associated to a finite
irreducible set of matrices is unique up to multiplication by a scalar, and
show that in certain cases these conditions are also persistent under small
perturbations.
