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

TL;DR

This paper introduces a new ergodic-theoretic framework called Mather sets for analyzing sequences of matrices that achieve maximum exponential growth, with applications to joint spectral radii and stability of linear systems.

Contribution

It develops a structure theorem for Mather sets of matrix sequences, linking ergodic theory with spectral radius and stability analysis, extending existing theorems in dynamical systems.

Findings

Characterization of Mather sets for matrix sequences

Application to joint spectral radius estimation

Insights into stability of linear inclusions

Abstract

The joint spectral radius of a compact set of d-times-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 semi-uniform 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.