On the Finiteness Property for Rational Matrices

Abstract

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This "finiteness conjecture" is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2 × 2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.

On a devil’s staircase associated to the joint spectral radii of a family of pairs of matrices

“…Despite having such a fast convergent infinite product for α γ , we still cannot use it to claim that α γ is irrational if γ is irrational. Such a result would show that the family (1.1) does not contain a counterexample to the rational finiteness conjecture (see [17] for more detail). Remark 8.10.…”

“…The existence of pairs of 2 × 2 matrices which do not satisfy the finiteness property was subsequently established by T. Bousch and J. Mairesse [4], with additional proofs being given later by V. Blondel, J. Theys and A. Vladimirov [3] and V. Kozyakin [19]. The finiteness property continues to be the subject of research investigation: some sufficient conditions for the finiteness property have been given in [6,7,8,17], and in a recent preprint N. Guglielmi and V. Protasov have given an algorithm for the rigorous verification of the finiteness property for real matrices [12].…”

On Complexity of Lyapunov Functions for Switched Linear Systems

“…In particular, the last reference provided the first explicit counterexample only recently. It is currently not known whether all sets of matrices with rational entries satisfy the finiteness property (Jungers and Blondel [2008a]). Gurvits [1992] shows that if the set of matrices admits a polytopic Lyapunov function, then the finiteness property holds.…”

“…More precisely, we have the following theorem: Theorem 9. (Jungers and Blondel [2008b]) Given a set of m nonnegative rational matrices Σ, it is possible to build a set of m binary matrices Σ (possibly of larger dimension), together with a natural number K such that for any product A = A i1 . .…”

The finiteness conjecture holds in SL(2,Z>=0)^2

“…The complexity of the matter already appears in the simplest setting, namely sets Σ containing just two 2 × 2 matrices. Indeed, such sets appear in the literature both as finiteness counterexamples [1], [10], [12], [19], as well as families of finiteness examples [15], [5], [17].…”