On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrodinger operator.

In this paper we study the Lyapunov exponent beta(E) for the one-dimensional Schrodinger operator with a quasi-periodic potential. Let Gamma subset of R-k be the set of frequency vectors whose components are rationally independent. Let 0 less than or equal to r < 1, and consider the complement in Gamma x C-r(T-k) of the set D where exponential dichotomy holds. We show that beta = 0 is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.