Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms
For researchers in matrix theory and dynamical systems, this work advances understanding of the stability of the finiteness property and uniqueness of Barabanov norms, though it is incremental.
The authors provide a condition ensuring that the finiteness property (maximal growth rate realized by a periodic product) holds for a finite irreducible set of matrices and all nearby sets, and prove conditions for uniqueness of Barabanov norms, showing persistence under perturbations.
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.