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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 ofmatrices to the spectral radii of finite products of those matrices. The proofrests on a theorem asserting the existence of a continuous invariant splittingfor certain matrix cocycles defined over a minimal homeomorphism and having theproperty that all forward products are uniformly bounded.

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