Mather sets for sequences of matrices and applications to the study of joint spectral radii

Abstract

The joint spectral radius of a compact set of d×d matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article, we investigate the ergodic‐theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences, which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semiuniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix.

Continuity properties of the lower spectral radius

“…, k} N with respect to which A in · · · A i1 1/n →ˇ (A) almost everywhere. This contrasts sharply with the situation for the upper spectral radius, where an ergodic measure with the analogous property always exists [45]. Since we make no use of ergodic theory in this article, it would be digressive for us to introduce the concepts required to prove this statement.…”

Datko-Pazy conditions for nonuniform exponential stability

“…It follows from (33) and [18,Theorem A.3.] that lim t→∞ 1 t log max q∈M A(q, t) < 0, which immediately implies (32).…”

“…We shall now show that Theorem 1 when combined with the result obtained by Morris [18] (building on the earlier work of Schreiber [30] and Sturman and Stark [32]) gives also new conditions for uniform exponential stability of continuous cocycles, i.e. of cocycles with the property that A : M → B(X) is a continuous map.…”

“…and (19) that there exists a measurable function T : M → (0, ∞) satisfying (18) and such that C(q) ≤ T (q) for µ-a.e. q ∈ M .…”

“…Here we show that assumptions in [25] can be considerably relaxed by assuming that those conditions hold for every point that belongs to the subset E of M , which is as large as M from the measure-theoretic point of view but can be considerably smaller from let's say dimension theory point of view (see the discussion after Theorem 3 for more details). This is achieved by using useful results of Morris [18] on the so-called subadditive ergodic optimisation.…”