A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

TL;DR

This paper employs ergodic theory to establish a rapidly-converging lower bound for the joint spectral radius, linking it to finite product spectral radii through invariant splitting techniques.

Contribution

It introduces a quantitative bound for the joint spectral radius using multiplicative ergodic theory and invariant splitting, advancing understanding of matrix cocycles.

Findings

Established a new lower bound for the joint spectral radius

Connected spectral radii of finite products to the joint spectral radius

Proved existence of a continuous invariant splitting under certain conditions

Abstract

We use ergodic theory to prove a quantitative version of a theorem of M. A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a theorem asserting the existence of a continuous invariant splitting for certain matrix cocycles defined over a minimal homeomorphism and having the property that all forward products are uniformly bounded.