  We first study the discrete Schr\"odinger equations with analytic potentials
given by a class of transformations. It is shown that if the coupling number is
large, then its logarithm equals approximately to the Lyapunov exponents. When
the transformation becomes the skew-shift, we prove that the Lyapunov exponent
is week H\"older continuous, and the spectrum satisfies Anderson Localization
and contains large intervals. Moreover, all of these conclusions are
non-perturbative.
