  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.
