Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

TL;DR

This paper establishes conditions ensuring the finiteness property and the uniqueness of Barabanov norms for finite irreducible matrix sets, and demonstrates their stability under small perturbations, advancing understanding of matrix product growth behavior.

Contribution

It provides new criteria for the stability of the finiteness property and Barabanov norm uniqueness under perturbations for finite irreducible matrix sets.

Findings

Finiteness property persists under certain conditions for nearby matrix sets.

Uniqueness of Barabanov norms is guaranteed and stable under small perturbations.

Conditions for stability are explicitly characterized for irreducible matrix sets.

Abstract

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.