Abstract: We used the methods such as the analytical approximation, the subharmonic function, the Birkhoff ergodic theorem, the large deviation theorem and the avalanche principle to study the regularity of the Lyapunov exponent corresponding to  the quasi-periodic Jacobi operator model. Under the condition that the potential energy is a non-degenerate Gevrey function, if the frequency is a strong Diophantine  number and the coefficient of potential energy is  large enough, we obtain the results that  Lyapunov exponent is positive and continuous. Thus, the  result of the Schrodinger cocycle in SL(2,R) is extended to the Jacobi cocycle  in GL(2,R).

Key words: quasi-periodic Jacobi operator, non-degenerate Gevrey function, Lyapunov exponent, continuity, positivity