Positive Lyapunov exponent for a class of quasi-periodic cocycles

Young [ 17 ] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrodinger cocycles by Zhang [ 18 ]. However, her result cannot be applied to the Schrodinger cocycles with the potential \begin{document}$ v = \cos(4\pi x)+w( x) $\end{document} , where \begin{document}$ w\in C^2(\mathbb R/\mathbb Z,\mathbb R) $\end{document} is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.


1.
Introduction. Positivity of Lyapunov exponent (short for LE) has attracted great attention in the study of quasi-periodic Schrödinger operators (cocycles). It is often taken as an implicit definition of localization in physics literature, and the LE is often called the inverse localization length. Although positivity of LE does not imply localization in the precise mathematical definition, it is usually treated as a precondition for localization. For example, Bourgain-Goldstein [5] showed Anderson Localization for the analytic quasi-periodic Schrödinger operators provided positivity of LE. Positivity of LE is also closely related to the spectral properties of the corresponding operators. It was proved by Ishii [10] and Pastur [13] that if the LE is positive for all energies, then there is no absolutely continuous component in the spectrum.
Proving positivity of LE is one of the central topics in this field. The most intensively studied cases are the quasi-periodic Schrödinger cocycles with real analytic potentials. A first proof is given by Herman [9] for the operators with trigonometric polynomial potentials by applying subharmonicity. Sorets-Spencer [15] further developed Herman's technique to prove positivity of LE for one-frequency nonconstant real analytic potentials with large disorders. This result was later extended to the Diophantine multi-frequency case by Bourgain-Goldstein [5] and any rational independent multi-frequency case by Bourgain [4]. It is worth to mention that Zhang [18] gave a different proof of the result in [15] based on the techniques in [1] and [17].
Besides analytic cases, positivity of LE also holds for some Gevrey classes of potentials and some finitely smooth potentials. The former is given with some strong Diophantine conditions by Elliason [7] and Klein [11]. For the so-called C 2 cos-type potentials, the positivity of LE (and even Anderson Localization) with 1362 JINHAO LIANG Diophantine frequencies was proved by Sinai [14] and Fröhlich-Spencer-Wittwer [8]. Similar results were obtained by different methods in [3] and [16]. The positivity of LE for some C 3 potentials was also established in [6] by excluding a positive measure of frequencies.
To prove the positivity of LE for smooth cocycles, an effective method was established without the nice properties of subharmonicity. In [17], Young developed a dynamical technique, which is close in spirit to the techniques in [2], to prove positivity of LE for a class of C 1 smooth SL(2, R) cocycles with Brjuno frequencies under a nonresonant condition. This result was applied to the Schrödinger cocycles with some C 1 potentials by excluding a small set of energies in [18]. Later, Wang-Zhang [16] developed a method from [17,18] to overcome the difficulties of resonances. And they gave the positivity of LE for the quasi-periodic Schrödinger cocycles with C 2 cos-type potentials, Diophantine frequencies and all energies.
In this paper, we aim to improve [17]'s result and then apply it to the quasiperiodic Schrödinger cocycles. We focus on a class of C 1 smooth cocylces in the following. For C, λ ≥ 1, we let A C,λ = {A ∈ C 1 (R/Z, SL(2, R)) : where C is a constant and λ is sufficiently large. Let A ∈ A C,λ . The map defines a family of dynamic systems on R/Z × R 2 , called a cocycle and denoted for simplicity by (α, A). The nth iteration of the cocycle is denoted by This limit always exists by Kingman's subadditive ergodic theorem. By the polar decomposition, we can rewrite A as where R θ is a rotation by θ and u, s ∈ RP 1 . Now we are ready to introduce the result in [17].
Though Theorem 1.1 is a bit different from Theorem 2 in [17], it is not hard to see that they are equivalent. Theorem 1.1 proves positivity of LE for a large class of C 1 smooth SL(2, R) cocycles. But since it needs A to be uniformly large, one cannot directly apply it to the Schrödinger cocycles. Zhang [18] found a conjugation of the Schrödinger cocycles to make the norm of the matrix sufficiently large, and then successfully applied Theorem 1.1 to the Schrödinger cocycles with some C 1 smooth potentials. In fact, he proved and has finitely many critical points, and for Then for each Brjuno frequency α, we have lim λ→∞ Leb( (λ)) = Leb(v(R/Z)).
Positivity of LE is established in Theorem 1.2 for the Schrödinger cocycle with a large subset of the potentials in C 1 (R/Z, R). However, the theorem cannot be applied to the simple potential: v(x) = cos(4πx) + w(x) with small perturbation w ∈ C 2 (R/Z, R). The reason that Theorem 1.2 cannot be applied is that the corresponding cocycle does not satisfy the Condition (T2) in Theorem 1.1. Hence this condition seems unessential. In fact, Young [17] used Condition (T2) to exclude all the energies with which resonances may happen, and to prove positivity of LE with no resonances. But the occurrence of resonances does not mean the vanishing of LE. It was proved in [16] that positivity of LE holds with some particular resonances, which are produced by only two points. This idea enlightens us to weaken Condition (T2).
In this paper, we will prove the positivity of LE with the conditions weaker than (T2). These conditions permit us to deal with some special resonances as in [16]. Here is our main theorem. 1], and α is a Diophantine number, i.e. ∃ γ > 0 and τ > 1 such that where x = min{|x − n| : n ∈ Z}. If g(x, t) satisfies Condition (T0), (T1) and one of the following conditions.
And for t outside a finite set, (∂g/∂t)/(∂g/∂x) takes the same value at no more than two points in C; (T4): For almost every t, (∂g/∂t)/(∂g/∂x) takes different values at different points in C.
(1.2) Remark 1.4. The frequency in Theorem 1.3 is not necessary to be Diophantine. One may use the technique in [12] to obtain the result for the Liouvillean frequency, which is defined as the irrational number satisfying lim sup n→∞ q −1 n ln q n+1 < ∞.
(Clearly, a Brjuno number is Liouvillean.) However, we still use the Diophantine condition here to simplify the proof. Theorem 1.3 can be applied to more cocycles than Theorem 1.1. The most important example is for the Schrödinger cocycles. Theorem 1.5. Let (α, A) be a Schrödinger cocycle as stated in Theorem 1.2. Assume that v is C 1 smooth and has finitely many critical points, and for t ∈ v(R/Z) outside a finite set, v (x) takes the same value at most two points Remark 1.6. For the Schrödinger cocycles, we only need x → g(x, t) to be C 1 smooth, instead of C 2 smooth for general cocycles. This will be illustrated in Section 5. Theorem 1.5 can be applied to a larger subset of the C 1 smooth potentials. Here we give an example that Theorem 1.2 does not hold but Theorem 1.5 holds. Let F be a subset of C 1 (R/Z, R) satisfying the following properties.
• v ∈ F has only four critical points.
As another example, we will show the positivity of LE for the analytic quasiperiodic Schrödinger cocycles. Although our result is weaker than [15] and [18], it gives an another way to deal with the analytic case. The rest of this paper is organized as follows. In Section 2 we will present some technical computations of product of two matrices.In Section 3 we will establish an induction theorem to deal with the resonances produced by only two critical points. In Section 4 we will apply the induction theorem to prove positivity of LE. In Section 5 we will estimate the measure of the parameter for which the resonances occur between only two critical points, and then finish the proof of Theorem 1.3, Theorem 1.5 and Corollary 1.8.

2.
Estimates for the product of the matrices. In this section, we consider the product of two SL(2, R) matrices. We will compute the angle between two matrices to obtain the norm of the product. Most of the lemmas are proved in [16].
Let A ∈ SL(2, R) satisfy A > 1. Define the map s, u : SL(2, R) → RP 1 := R/(πZ), such that s(A) is the most contraction direction of A and u(A) = s(A −1 ). Then for A ∈ SL(2, R) with A > 1, we have the polar decomposition where s, u ∈ [0, 2π) are some suitable choices of angles corresponding to the directions s(A), u(A) ∈ RP 1 . For convenience, we also use s(A) and u(A) for s and u.
Let I be an interval and x ∈ I. Assume that We say |s(E 2 ) − u(E 1 )| is the angle between E 2 and E 1 . If E 1 , E 2 1, then it is not hard to see that E 2 E 1 approximately equals to C · E 2 · E 1 (C > 0) unless the angle tends to 0. We say it is nonresonant if Otherwise, we say it is resonant. The following lemma gives a method to compute Type III: Type III: f 1 (c1)f 2 (c2) < 0 Figure 2. Graphs of the angle functions As we see in Lemma 2.1, the angle function θ(x) = s 2 (x) − u 1 (x) takes an important role in the product of two matrices. In order to study the function, we introduce some types of the angle functions.
Definition 2.1. Let B(x, r) ⊂ R/Z be the ball centered around x ∈ R/Z with a radius of r. For a connected interval J ⊂ R/Z and a constant 0 < a ≤ 1, let aJ be the subinterval of J with the same center and whose length is a|J|.
. We define the following types of the angle functions, see Figure 2.
• f is of type I. If we have the following.
f C 2 < C and f (x) = 0 has only one solution, say x 0 , which is contained in I 3 ; -df dx = 0 has at most one solution on I while df dx > r 2 for all x ∈ B(x 0 , r 2 ); Here f 1 and f 2 are of type I.
Remark 2.5. The type III function in [16] always satisfies In our setting, we also consider the case with Figure 2. In the induction process, we will prove that the sign of Remark 2.6. In [16], the authors also consider the so-called "type II" angle function, which behaves as a quadratic function. However, in our setting the energies which lead to the case with type II angle function have arbitrary small measure.
Since our result concerns about the measure estimate, excluding a small set is allowed. For simplicity, we do not consider the "type II" angle function.
We care about the non-degenerate property of the angle functions. The function of type I behaves as an affine function and is non-degenerate near the zeros. For the functions of type III, they can be divided into two parts: arctan l 2 tan f 1 (x) − π to be neglected and therefore f (x) approximately equals to f 2 (x). If x varies near the zero of f 1 , the first part has a drastic change from −π to 0. Then the number of zeros depends on the sign of f 1 (c 1 )f 2 (c 2 ). If f 1 (c 1 )f 2 (c 2 ) > 0, then f is monotone in I and has exactly two zeros. If f 1 (c 1 )f 2 (c 2 ) < 0, then f is not monotone and may has no zero, one zero or two zeros. See Figure 2. In the appendix, Lemma A.1 gives a careful description of the functions of type III. And Lemma A.3 shows that the type III functions are also non-degenerate. 3. The induction process. In this section we will establish an induction theorem similar to [16] to deal with the resonances, which are produced by only two critical points. More precisely, we will iterate to show A rn (x) has a large lower bound with r n → ∞, provided some assumptions on the critical points and the coupling constant λ sufficiently large. Moreover, the largeness of λ does not depend on the choice of t.
We first explain the idea of the induction process. To estimate A rn (x) with some suitable r n , we start with the product Before repeatedly applying Lemma 2.1 to the above product, we have to confirm whether the angle between A(x + jα) and A(x + (j − 1)α) is not too small. The non-degenerate property of the angle function is needed here. And we require α be Diophantine and x locate near a zero of the angle function such that r 1 can be large enough. Then Lemma 2.1 gives the estimates of A r1 (x) , s(A r1 (x)) and u(A r1 (x)). Proceed to the next step, we consider the product We will use the estimates of A r1 (x) , s(A r1 (x)) and u(A r1 (x)) to verify the conditions of Lemma 2.1 and again repeatedly apply Lemma 2.1 to get the lower bound of A r2 (x) . Inductively, the estimates of A rn (x) , s(A rn (x)) and u(A rn (x)) can be obtained in the same way.
In the process of estimating the angles, we are afraid that x + jα locates near one zero of the angle function provided that j is small and x locates near another zero. More precisely, if x locates near one zero, then the trajectory {x + jα : 0 < j < q n } (p n /q n is the continued fraction approximant of α, see Appendix B for more properties) may goes into the neighbourhoods of other zeros k times. We say this case is a resonant case. If k > 1, then it is hard to get a desired lower bound of A qn (x) since we cannot verify the condition "e 1 ≤ e β 2 " in Lemma 2.1. Moreover, the resonant case causes a great change of the angle function (see Resonant case in Lemma 2.1) and brings a lot of troubles in the estimation of the angles. Due to the complexity of the resonance case, Young [17] excluded all the energies which may lead to the resonant cases. However, not all the resonances are hard to be dealt with. In [16], the authors found that if the original angle function has only two zeros, then the trajectory {x + jα : 0 < j < q n } goes into the neighbourhoods of other zeros at most k = 1 time. And then one can use Lemma 2.1 to deal with the resonance with k = 1 by verifying the conditions with the Diophantine condition on the frequency. In our setting, the angle function may have many zeros and k may larger than 1. However, we can exclude all the "bad" energies which leads to k > 1 such that we only need to consider the resonance with k = 1. In this section, we are going to establish an induction theorem, which is similar to the induction theorem in [16], to deal with the resonance with k = 1. In Section 5, the estimate of the measure of the "bad" energies will be obtained. Now we go into the detail. Condition (T0) says that #C(t) is finite. Let Hence g is a type I function in each I N,j . Note that N can be chosen independent of t.
Consider the sequence {λ n : n ≥ N } defined by Since α is Diophantine and N is sufficiently large, it is easy to see that λ n decreases to some λ ∞ > 0 and η n increases to some η ∞ . And we have λ ∞ > λ 1− ε 2 and η ∞ < ε. Fix any t ∈ [0, 1] below. Let We define the initial angle function g N as and the initial critical points c N,j as For n ≥ N , we inductively define the (n)th critical interval, the (n)th return time, the (n + 1)th angle function and the (n + 1)th critical points as the following.
• The critical interval I n,j centers at the critical point c n,j , with radius of q −3τ n on R/Z. • The return time r ± n,j (x) : I n,j → Z + is the first return time to I n after q n , where r + n,j (x) is the forward return time and r − n,j (x) backward. That is r ± n,j (x) = min{l : x ± lα ∈ I n , l ≥ q n }, x ∈ I n,j . Let r ± n,j = min x∈In,j r ± n,j (x) and r n,j = min{r + n,j , r − n,j }. And let r N −1,j = 0. • The angle function g n+1 : . Moreover, the critical point c n+1,j is the minimal point of the angle function g n+1 as the following We assume the following properties about C (n) .
Now we are ready to give the induction theorem.
Theorem 3.1. Let ε be small enough. Let λ be large enough such that Assume that C (ñ) satisfies Assumption (ñ) for N ≤ñ ≤ n + 1. We also assume that the following properties holds in the (n)th step.
• We have the estimates of the norm of A ±r ± n,i (x) and its derivatives. For • Depending on the positions of the critical intervals (points), it is divided into the nonresonant case and the resonant case, which leads to different types of the angle functions.
-In the nonresonant case, the angle function g n+1 is of type I in I n,i . And we have Moreover, it satisfies g n+1 C 2 ≤ C. -In the resonant case, the angle function g n+1 is of type III in I n,i (i = 1, 2). And it satisfies g n+1 C 2 ≤ Cλ 5qn . Moreover, the function g n+1 satisfies the following properties: * If d dx g n (c n,i ) d dx g n (c n,j ) > 0, then g n+1 is monotone and has exactly 2 zeros in I n,i . The same is true for g n+1 in I n,j . And we also have dx g n (c n,j ) ≤ 0, then g n+1 may have 0, 1 or 2 zeros in I n,i . And we also have Furthermore, g n+1 has one more minimal point c n+1,j on I n+1,i such that g n+1 (c n+1,j ) = g n+1 (c n+1,i ), and g n+1 has one more minimal point c n+1,i on I n+1,j such that g n+1 (c n+1,i ) = g n+1 (c n+1,j ). And it holds that * if |g n (c n,i )|, |g n (c n,j )| > Cλ − 1 10 rn,i , then Then we can push the induction to the (n + 1)th step with the above properties by replacing n by n + 1.
Remark 3.2. It can be seen in the proof that the choice of λ 0 is independent of t. Remark 3.3. Instead of r ± n,i (x), we used r ± n,i and r n,i . The differences between the usage of r ± n,i (x), r ± n,i and r n,i are negligible. In fact, by Lemma 2.1, it is easy to see that these differences produce errors which are smaller than λ − 3 2 rn,i and are clearly not important in all the necessary estimates. Hence, we will not distinguish the differences among r ± n,i (x), r ± n,i and r n,i .
The proof of Theorem 3.1 is similar to the proof of Theorem 3 in [16]. Although it is important, it is quite lengthy. To keep the paper compact, we put the proof into the Appendix C.
4. Estimating the Lyapunov exponent. Proof. Our idea is to construct an exceptional set B ⊆ S 1 with small measure, and to estimate the lower bound of A i (x) outside B by using the induction theorem.
Let q n1 = q N and q n k+1 = min{q l : q l > q 4τ n k }. Let i = q 2 n k+1 and provided sufficiently large N . Let i 0 be the first time such that x + i 0 α ∈ I n k . By Lemma B.2, we have i 0 ≤ q n k+1 i. Then we can assume that x ∈ I n k . Similarly, we can also assume that x + iα ∈ I n k . Since x ∈ B n k , the trajectory {x + jα : 1 ≤ j ≤ i} has not entered I n k+1 , but has entered I n k . Let 0 = j 0 < j 1 < · · · < j p = i be the return times to I n k such that j l+1 − j l ≥ q n k . Since the trajectory has not entered I n k+1 , then dist(x + j l α, C (n k+1 ) ) > q −3τ n k+1 . By the definition of n k+1 , we have q n k+1 ≤ q τ n k+1 −1 ≤ q 4τ 2 n k .

JINHAO LIANG
In the nonresonant case, since there are no extra zeros of g n k +1 , then In the resonant case, we are worried about the possibility that However, either |c n k ,2 − c n k ,1 | < λ − 3 4 qn k so that |x + j l α − c n k ,1 | < Cq −3τ n k+1 , contradicting that the trajectory has not entered I n k+1 . Or for some |k| < q n k , This implies |x + j l α + kα − c n k ,2 | < Cq −3τ n k+1 . This also contradicts that the trajectory has not entered I n k+1 . It concludes by Then by Lemma 2.1, we have where we use p · q n k ≤ i and ln λ > 1 in the second inequality. Then ln λ, i = q 2 n k+1 + q n k+1 , for all k ≥ 1 and x ∈ (R/Z)\ k≥1 B n k . By Kingman's Subadditive Ergodic Theorem, the LE always exists and it holds that

5.
Estimate the measure of the good parameter. We will mainly follow the idea in [17] to estimate the measure of the good parameter. However, since the elements in C(t) do not need to take different derivatives with respect to t, the derivative of the difference of two critical points has no uniformly lower bound. In this section, we will obtain a lower bound depending on the contraction of every connected component of t. By controlling λ to be sufficiently large, we can ensure that the contraction is small enough. And then we can use the lower bound to exclude the bad parameter.
We will divide the proof into two cases: inside T and outside T .
Choose N sufficiently large such that |J|.

Now we define
Assumption (n ). ∀ a, b ∈ C (n) , for 0 < |k| < q n , If C (n) satisfies Assumption (n ) for all n ≥ N , then C (n) satisfies Assumption (n) for all n ≥ N . By Theorem 3.1, then there exists a λ J such that for λ > λ J the induction process in Section 3 applies. In fact, each step in the induction process is nonresonant. Hence we obtain the positivity of LE by Theorem 4.1. It remains to estimate the measure of t with which C (n) (t) satisfies Assumption (n ) for all n ≥ N .
Lemma 5.1. There exist γ 1 , γ 2 > 0 such that the following holds. (In fact, Let ω be any subinterval of J on which C (n) is defined and r ± n−1,j is independent of t for every j. Let a = b ∈ C, and let a (n) (t), b (n) (t) ∈ C (n) (t) denote the corresponding critical points. Then for t ∈ ω, Proof. This lemma follows easily from lemma 2.1 and the induction theorem.
Moving on to the next step, we let P N +1 be a refinement of P N |∆ N , subdividing ∆ N into intervals of length ≈ q −3τ N +1 . We define For each ω ⊂ ∆ N +1 ,r ± N +1,j and C (N +2) are also defined similarly. This process gives a decreasing sequence of sets such that for any t ∈ ∆ n , C (l) satisfies Assumption (l ) for N ≤ l ≤ n. Moreover, there is an increasing sequence of partitions {P n }, defined on ∆ n , such that the definition of C (n+1) is consistent on each element of P n . Hence Lemma 5.1 applies.
Consider a, b ∈ C. Define the function τ N : For n > N , let τ n : Lemma 5.2. τ n : ∆ n−1 → R is at most 2 n−N to 1.
Proof. Let ω ∈ P n−1 . If ω , ω ⊂ ω are two non-adjacent elements of P n |∆ n , ω to the left of ω , then This comes from By Lemma 5.1 and Lemma 5.2, And thus Let M be a set of finitely many Js such that Let c i , c j ∈ C. We consider Let F i,j = (0, 1)\E i,j . Note that F i,j is an open set. Then {ĩ,j} ={i,j} Fĩ ,j consists of a sequence of intervals. Let J be one of those intervals. Let t ∈ (1 − ε 100 )J. Then there exists γ 0 = γ 0 (ε) such that for any a, b ∈ C\{c i , c j }, Choose N sufficiently large such that We define If t ∈ E i,j and C (n) (t) satisfies Assumption (n i,j ) for all n ≥ N , then C (n) satisfies Assumption (n) for all n ≥ N by Condition (T3). By Theorem 3.1, there exists a λ J such that for λ > λ J the induction process in Section 3 applies. By Theorem 4.1 we obtain the positivity of LE. It remains to estimate the measure of such t.
Similar to in the previous subsection, we can define a sequence of ∆ n such that and for any t ∈ ∆ n , C (l) satisfies Assumption (l i,j ) for N ≤ l ≤ n. Moreover, there is an increasing sequence of partitions {P n }, defined on ∆ n , such that the definition of C (n+1) is consistent on each element of P n . Hence Lemma 5.1 applies. Using the same argument in the previous subsection, we can obtain Let M be a set of finitely many Js such that Then it holds that Leb(∪ J∈M ∩ n≥N ∆ n (J)) ≥ (1 − ε)Leb(∩ {ĩ,j} ={i,j} Fĩ ,j ).
Let λ > λ i,j := max{λ J : J ∈ M }. If t ∈ E i,j ∩ G i,j , then C (n) satisfies Assumption (n i,j ) for all n ≥ N . And hence C (n) satisfies Assumption (n) by Condition (T3). By Theorem 4.1 we obtain the positivity of LE. Therefore, we have Similarly, we consider ∩ {i,j}∈K E i,j for arbitrary nonempty subset K ⊂ {{i, j} : 1 ≤ i, j ≤ m, i = j}. Note {i,j} / ∈K F i,j consists of a sequence of open intervals. As before, there exists λ K sufficiently large such that if λ > λ K , the following holds. There exists G K satisfying G K ⊂ {i,j} / ∈K F i,j such that for t ∈ G K , the corresponding cocycle satisfies Assumption (n K ). If {a, b} ∈ {{a, b} : a, b ∈ C (n) , a = b}\{{c n,i , c n,j } : {i, j} ∈ K},

Moreover, we have
). Note that Condition (T3) implies that for t ∈ G K , the cocycle satisfies Assumption (n). By Theorem 4.1, we obtain . Take summations on both sides of (5.2) This implies (1.2). If Condition (T4) holds, then for almost every t, each element in C has different derivatives with respect to t. In other words, Leb(T ) = 0. By (5.1), we have (1.2). Note that in this case, we do not deal with the resonant case in Section 3. Hence we do not need the cocycle to be C 2 smooth.

5.4.
Proof of Theorem 1.5. Now we consider the Schrödinger cocycle with the form stated in Theorem 1.2. Assume that v is C 1 smooth and has finitely many critical points. By Appendix A.1 in [16] or Section 5.3 in [18], we can choose where χ is C 1 smooth on x and t, and is continuous on λ. And it holds that for any x ∈ R/Z and t ∈ v(R/Z), then such t has small measure. In fact, since v has finitely many critical points, {v(x) : |v (x)| < ζ 0 } is contained in finitely many intervals, which are extremely small for ζ 0 small enough. Let λ satisfy λ > C 1 ζ −1 0 . Then the measure of v(x) satisfies the second equality must be contained in finitely many small intervals. Then by the first equality, t must be contained in finitely many small intervals. Therefore Condition (T0) and Condition (T1) hold for t outside finitely many small intervals. Since our result concerns about the measure estimate, excluding a small set is allowed. Now we are going to verify Condition (T3). As above, we only show it for t outside a set of small measure. For x satisfying t − v(x) = 0, we have a solution c = c(t) of g N (c, t) = 0 near x for large λ. Let c = c j (t) be the solution of g N (c, t) = 0 and Recall the condition that v (x) takes the same value at most two points in as ζ goes to zero. In fact, since as ζ goes to zero. To obtain Condition (T3), we need the following lemma.
Then for λ sufficiently large depending on ζ and ζ 0 , Then for λ sufficiently large depending on ζ, Similar estimate holds for j. Then We also have c i (t)(v (c i ) + ∂ x χ) = −1 − ∂ t χ, which implies that c i (t)v (c i ) = 1 + O(λ −1 ) for large λ depending on ζ 0 . Then the lemma is proved for sufficiently large λ depending on ζ and ζ 0 .
Combining this lemma and (5.4), we obtain that if λ sufficiently large, then Condition (T3) holds for t outside a set of small measure.
What remains to do is to show we only need v to be C 1 smooth in the induction process. For any c i (t) ∈ C(t), we have g N (c i (t), t) = 0. Take derivative with respect to t on both sides, we get for λ sufficiently large. Hence it follows that ∂ x g N (c i (t), t) and c i (t) have different signs. If c i (t) · c j (t) > 0, then Since is also an open set. Now we follow the procedure in Subsection 5.2 by replacing {i,j} / ∈K F i,j by (5.6). Note that for each t ∈ E i,j ∩ ( {i,j} / ∈K F i,j ), we have (5.5). Then we can inductively redefine C (n) with ∂ x g n (c n,i (t), t) · ∂ x g n (c n,j (t), t) > 0. (5.7) By Remark 2.4 and Remark A.2, we only need the C 1 estimates in the induction process. Then we can follow Subsection 5.2 to prove Theorem 1.5.

5.5.
Proof of corollary 1.8. Now we consider the Schrödinger cocycle with the form stated in Theorem 1.2. Assume that v is real analytic on {x : | x| < h}. We will show that the cocycle satisfies Condition (T4), i.e. Leb(T ) = 0. By Appendix A.1 in [16] or Section 5.3 in [18], it is not hard to see that where χ is analytic on x and t, and is continuous on λ > 0. Moreover, it holds that for any x ∈ {| x| < h} and t ∈ v(R/Z), where C 2 does not depend on x, t and λ. Due the analyticity of g N on x, it is easy to see that Condition(T0) holds and Condition (T1) holds except for finitely many t. Moreover, since v is analytic, {x : v (x) = 0} is a finite set. There exist an open interval J and a positive number ζ such that for each t ∈ J, ∃x satisfying t = v(x) and v (x) > ζ > 0. Let λ be large enough such that C 2 λ −1 < ζ. By the Implicit Function Theorem, we can obtain x = x(t) is analytic with respect to t. Hence for c ∈ C, c(t) is analytic on J. Moveover, since c(t) is monotone on J, c(J) is also an open interval. Suppose Leb(T ) > 0. Note that Condition (T0) gives that #C is finite. Then there exists i, j such that Leb(E i,j ) > 0. Hence E i,j has an accumulation point. Since c i (t) − c j (t) is analytic, we get that c i (t) − c j (t) = 0 for all t ∈ J. Then there exists a constant T 1 satisfying 0 < T 1 < 1 and c i (t) = c j (t) + T 1 , t ∈ J.

Now we consider the function
We will prove h(x) ≡ 0 on x ∈ c j (J). If not, then h has finitely many zeros in {x : x ∈ c j (J), | x| < h}. Denote the set of the zeros by X. Let K be a compact subset of {x : x ∈ c j (J), | x| < h}. By Lojasiewicz inequality, there exist positive constants γ and C 3 such that Let f (x) = χ(x, t, λ) − χ(x + T 1 , t, λ). Then we have This leads to contradiction since the left hand side of the above inequality is a positive constant and the right hand side can be extremely small by choosing λ sufficiently large. Hence we have h(x) ≡ 0 on x ∈ c j (J). By the analyticity of h, it holds that v(x + T 1 ) ≡ v(x), x ∈ R/Z. This implies that v has a period T 1 .
Note that v cannot have infinitely many periods which are smaller than 1. Otherwise, v must be constant. Assume that v has a minimal period T 0 > 0. Let w(x) := v(T 0 x). Then w has a minimal period 1. Let We consider the cocycle (α, B). Then the corresponding T satisfies Leb(T ) = 0. In fact, if Leb(T ) > 0, then by the above discussion we get that w has another period 0 < T 2 < 1. Then v has a period T 0 · T 2 , contradicting the minimality of T 0 . Hence the cocycle (α, B) satisfies Condition (T4). And by Theorem 1.3, we have (1.2) for (α, B). Moreover, direct computation shows that L(α, B) = L(α, A).
This finishes the proof of Corollary 1.8.
if 0 ≤ |c 1 − c 2 | < η 0 , then f has no zeros and Remark A.2. Note that the angle function f discussed above is C 2 smooth. However, if f ∈ C 1 (I, RP 1 ), then we can also define type I function and type III function as in Definition 2.1 by removing the estimate of |f | and |l |. In Lemma A.1, we need f to be C 2 smooth only in the case with f 1 (c 1 )f 2 (c 2 ) < 0. In other words, if f is C 1 smooth and f 1 (c 1 )f 2 (c 2 ) > 0, then Lemma A.1 also holds.
No matter which type of the angle function is, the angle function satisfies the following non-degenerate property. In case f is of type III, we further assume d := |x 1 − x 2 | < r 3 . Then for any 0 < r < r, we have that |f (x)| > cr 3 , for x / ∈ B (X, r ) .
For the case that f is of type III, we have the same estimate for Cl − 1 4 < r < r if d ≥ r 3 . Proof. Most of the results are proved in Corollary 3 in [16] except for the type III function f with f 1 (c 1 )f 2 (c 2 ) > 0. And this is obvious from Lemma A.1.
Appendix B: Arithmetic properties of the frequencies. Let {p n /q n } be the continued fraction approximant of α. It is well known that {p n /q n } satisfies 1 q n (q n+1 + q n ) ≤ α − p n q n ≤ 1 q n q n+1 . (B.1) Lemma B.1. Let α be a Diophantine number as in (1.1), c ∈ R/Z and x ∈ B(c, q −3τ n ) . Then for k satisfying 0 < |k| < q 2 n , we have x + kα ∈ B(c, q −3τ n ).
Since q −1 l < q −4τ n q −3τ n , there exists i 0 such that 0 < i 0 < q l and x + iα ∈ x + i 0 p l q l − 1 q l , x + i 0 p l q l + 1 q l ⊂ I.
This implies the result. Now we consider the nth resonant case. For x ∈ I n,i , there exists k such that 0 < |k | < q n and (I n,i + k α) ∩ I n,j = ∅. Without loss of generality, we assume that k > 0. By Diophantine condition, we get k = k . And hence 0 < k < q n−1 ≤ r + n−1,i . We consider the decomposition A r + n,i (x) = A r + n,i −r + n−1,i (x + r + n−1,i α)A r + n−1,i (x).
Using the same argument in the nth nonresonant case, one obtains Similarly, we can obtain (C.1) on I n,j . If |g n (c n,i )|, |g n (c n,j )| > Cλ − 1 10 rn,j , then by (C.1) |g n+1 (c n+1,i )|, |g n+1 (c n+1,j )| > Cλ − 1 10 rn,j . Moreover, since g n is a type III function, g n+1 is also a type III function. Then by Lemma A.1, g n+1 satisfies the properties stated in the induction theorem. Moreover, we also have Lemma C.2 in this case.
Until now we have finished the proof of Theorem 3.1.