A few days ago I attended a lecture given by Amie Wilkinson. She presented a proof of Furstenberg’s theorem on the Lyapunov exponents of random products of matrices in .
Let be a probability measure on , be the product measure on . Let be the shift map on , and be the projection. We consider the induced skew product on . The (largest) Lyapunov exponent of is defined to be the value such that for -a.e. .
To apply the ergodic theory, we first assume . Then is well defined. There are cases when :
(1) the generated group is compact;
(2) there exists a finite set of lines that is invariant for all .
Furstenberg proved that the above cover all cases with zero exponent:
for all other .