A Thouless formula and Aubry duality for long-range Schrödinger skew-products

In this paper, we study the dynamical properties of a class of ergodic linear skew-products which includes the linear skew-products defined by quasi-periodic Schrödinger operators and their duals, in Aubry sense, when the potential is a trigonometric polynomial. Notably, these linear skew-products preserve an adapted complex-symplectic structure. We prove a Thouless formula relating the sum of the positive Lyapunov exponents and the logarithmic potential associated with the density of states of the corresponding operator. In particular, for quasi-periodic Schrödinger operators and their duals, we prove an identity for the upper Lyapunov exponent of the skew-product and the sum of the positive Lyapunov exponents of their dual, which generalizes the well-known formula for the Almost Mathieu. We illustrate these identities with some numerical illustrations.