Lyapunov Functions and Cones

In this chapter we describe how one can use Lyapunov functions to show that a cocycle over a measure-preserving transformation has a nonvanishing Lyapunov exponent almost everywhere. Starting with the simpler case of a nonpositive Lyapunov function, we show that the existence of an eventually strict nonpositive Lyapunov function implies that the Lyapunov exponent is negative almost everywhere. Then we consider arbitrary Lyapunov functions and, analogously, we show that the existence of an eventually strict Lyapunov function implies that the Lyapunov exponent is nonzero almost everywhere. Finally, we briefly consider the case of a single sequence of matrices. For a certain class of sequences, we describe a criterion for a nonvanishing top Lyapunov exponent in terms of invariant cone families. We note that the theory is essentially different in the cases of a cocycle and of a single sequence of matrices, mainly because in the latter case one cannot use the powerful tools of ergodic theory.