  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.
