PROPERTIES OF LYAPUNOV EXPONENTS FOR QUASIPERODIC COCYCLES WITH SINGULARITIES

We consider the quasi-periodic cocycles (omega, A(x, E)) : (x, v) -> (x + omega A(x, E)v) with omega Diophantine. Let M-2(C) be a normed space endowed with the matrix norm, whose elements are the 2 x 2 matrices. Assume that A : X x epsilon -> M-2(C) is jointly continuous, depends analytically on x is an element of T and is Holder continuous in E is an element of epsilon, where epsilon is a compact metric space and T is the torus. We prove that if two Lyapunov exponents are distinct at one point E-0 is an element of epsilon, then these two Lyapunov exponents are Holder continuous at any E in a ball central at Eo. Moreover, we will give the expressions of the radius of this ball and the Holder exponents of the two Lyapunov exponents.