Uniform Positivity And H(?)lder Continuity Of The Lyapunov Exponent For A Class Of Smooth Quasi-periodic Schr(?)dinger Cocycles

We consider the one-dimensional discrete quasi-periodic Schr(?)dinger operator act-ing on l2(Z),H:l2(Z)? l2(Z)(H?,?,v,xu)n:= un+1 + un-1 + ?v(x + n?)un,where v is a C2 cos-type potential function on R/Z,? is an arbitrary Liouville frequen-cy,? is a coupling constant.We are going to prove that if ? is sufficiently large,then the Lyapunov exponent of the corresponding Schodinger cocycle is uniformly positive and H(?)lder continuous as a function of the energy.Moreover,we will show that the H(?)lder exponent is a uniform constant independent of ? and ?.In the first chapter,we will first introduce the background and the previous results in the literature.We then give the main results in detail.At last,we briefly introduce the main method used in the proof,and then explain the basic ideas of the proof.In the second chapter,we will give some basic knowledge,such as the relation between the Schr(?)dinger operators and Schodinger cocycles,the definition and the basic properties of the Lyapunov exponent,and the concept of the integral density of state.In the third chapter,we will introduce the large deviation theory and the avalanche principle,which are developed by Bourgain,Goldstein and Schlag.We also give a proof of the uniform positivity and the H(?)lder continuity of the Lyapunov exponent for the general analytic potentials by their methods.Next,we will introduce the techniques,developed by Young,Wang and Zhang,to study the iteration of the Schr(?)dinger cocy-cle.These methods will be used in the proof of our main results.In the last chapter,we will give the proof of the main results.We will introduce the properties of the fraction approximation of the general irrational numbers,and then carefully analyze the trajectory of the dynamics on the bottom space.Then,we follow the approach of Wang-Zhang[36]and establish the induction theorem for the Liouville frequencies,from which the lower bound of the Lyapunov exponent can be obtained.Next,basing on the induction process,we divide the trajectory into several segments and combine them together.The key point here is to classify all the possibilities of the resonance and deal with them separately.At last,we will get a refined large derivation theorem,from which the H(?)lder continuity of the Lyapunov exponent follows.