Lyapunov Exponents of Discrete Analytic Schrödinger Equations with Multivariable Gevrey Perturbation

In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You that the authors constructed the examples whose Lyapunov exponent is discontinuous in the potential with the C⁰ norm for non-analytic potentials. In this paper, we consider these operators for some Gevrey potential, which is an analytic function with a Gevrey small perturbation on the multi-dimensional torus. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous with the C⁰ norm in all parameters, such as the energy, the frequency and the potential, for two transformations, the multi-frequency shift and the skew shift. It is complementary to Wang-You’s result.