The Lyapunov Exponent Of Quasi-periodic Operators

This paper is a survey studying about the basic properties of the Lyapunov exponent of quasiperiodic operators with analytical potnetial. We introduce the result by J.Bourgain and SJitomirskaya that the Lyapunov exponent is continuous on parameter energy for 1-dimensional irrational frequencies. In the final, we also present a conjecture shows that the Lyapunov exponent may not depend on the frequency under the assumption of C2 potential.The whole paper contains four sections. First section introduces the large deviation theorem and avalanche principle. And the proof of the main result can be found in the second section. Then we discuss the conjecture mentioned above. In the last section all the proofs of the theorems are provided.