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

We study the Chirikov (standard) map at large coupling $λ\gg 1$, and prove that the Lyapounov exponent of the associated Schroedinger operator is of order $\log λ$ except for a set of energies of measure $\exp(-c λ^β)$ for some $1 < β< 2$. 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.