Properties of Lyapunov Exponents for Quasiperodic Cocycles with Singularities

We consider the quasi-periodic cocycles (ω, A(x, E)) : (x, v) 7→ (x+ω, A(x, E)v) with ω Diophantine. Let M2(C) be a normed space endowed with the matrix norm, whose elements are the 2 × 2 matrices. Assume that A : T × E → M2(C) is jointly continuous, depends analytically on x ∈ T and is Hölder continuous in E ∈ E , where E is a compact metric space and T is the torus. We prove that if two Lyapunov exponents are distinct at one point E0 ∈ E , then these two Lyapunov exponents are Hölder continuous at any E in a ball central at E0. Moreover, we will give the expressions of the radius of this ball and the Hölder exponents of the two Lyapunov exponents.