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

By Ian D. Morris. 2011

74 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/090715f300914ec37ab99af0e6f347233c0250a2