Bounds on the Lyapunov exponent via crude estimates on the density of states

We study the Chirikov (standard) map at large coupling $\lambda \gg 1$, and prove that the Lyapounov exponent of the associated Schroedinger operator is of order $\log \lambda$ except for a set of energies of measure $\exp(-c \lambda^\beta)$ for some $1 < \beta < 2$. We also prove a similar (sharp) lower bound on the Lyapunov exponent (outside a small exceptional set of energies) for a large family of ergodic Schroedinger operators, the prime example being the $d$-dimensional skew shift.