Harmonicity of the Lyapunov exponent for monotonic cocycles and a generalized Thouless formula

In the case of Schrödinger cocycles, by the so-called Thouless formula the harmonicity of the Lyapunov exponent is equivalent to uniform hyperbolicity. In this paper we consider monotonic cocycles and give a sufficient condition for the harmonicity of the Lyapunov exponent to be equivalent to uniform hyperbolicity. We then generalize the Thouless formula to more general cocycles.