ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL

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1 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL HELGE KRÜGER Abstract. I will prove that for most energies E and some frequencies α the Lyapunov exponent of the Schrödinger operator with potential V (n) = 2λ cos(2παn 2 ) has the behavior L(E) λ 2 as λ 0. This improves upon earlier results by Bourgain.. Introduction In this paper, I wish to analyze the skew-shift Schrödinger operator at small coupling. That is for λ > 0 small and α irrational the potential (.) V α,λ (n) = 2λ cos(2παn 2 ) and the associated Schrödinger operator H α,λ = +V α,λ. The Lyapunov exponent L α,λ (E) describes the maximal exponential growth of solutions of H α,λ u = Eu. We will give a definition in (2.4). See also [7] or [5] for background. It is known that for irrational α and λ > large enough, we have L α,λ (E) log λ. This can be shown by using Herman s subharmonicity trick from [], or under suitable additional assumptions on α or E by the more constructive methods of Bourgain [3], Bourgain, Goldstein, and Schlag [6], or myself [2]. My main goal is to improve the result of Bourgain from [] on positivity of the Lyapunov exponent for this model for small λ > 0, certain irrational α, and most energies E. I will prove the following quantitative version of Bourgain s result. Theorem.. Given δ > 0 and τ > 0. There are λ 0 = λ 0 (δ, τ) > 0 and γ 0 = γ 0 (δ, τ) > 0. For each 0 < λ < λ 0, there is α 0 (λ) > 0 such that for 0 < α < α 0 irrational, there exists a set E b = E b (λ, α) of measure (.2) E b τ such that for E ([ 2 + δ, δ] [δ, 2 δ]) \ E b (.3) L α,λ (E) γ 0 λ 2. The main improvement is the quantitative lower bound L α,λ (E) λ 2. This behavior is expected to hold for all irrational α, E ( 2, 2) \ {0}, and 0 < λ. However even positivity is unknown (see e.g. [4], [4]). In my earlier paper [2], I established a similar result for E ( 2, 2) \ {0}. However, I had to replace the Date: April 27, Mathematics Subject Classification. Primary 8Q0; Secondary 37D25. Key words and phrases. Lyapunov Exponents, Schrödinger Operators. H. K. was supported by NSF grant DMS and a Nettie S. Autrey Fellowship.

2 2 H. KRÜGER power 2 in (.) by a higher one, so that V (n) = 2λ cos(2παn k ) for an irrational α and large enough k depending on λ > 0. That E b doesn t converge to 0 as λ 0 comes from that I wish to ensure the behavior L α,λ (E) λ 2. One could also show for ε > 0 that L α,λ (E) λ 2+ε with E b λ ε and uniform constants. For this and a discussion of the size of α, see the end of Section 3. Combining the proof of the previous theorem with the results of Bougain, Goldstein, and Schlag in [6], I can furthermore show Theorem.2. Let λ and α satisfy the same conditions as in Theorem.. Assume furthermore that α satisfies a Diophantine condition (.4) dist(nα, Z) κ n t for n Z \ {0} and some fixed κ, t > 0. Then the set E b can be chosen to be a finite union of intervals. The problem with is that the methods from [6] are not directly applicable, since our initial condition only holds with low probability, and [6] needs probabilities extremely close to (depending on the initial estimate on the Lyapunov exponent). We will demonstrate how this can be achieved using our methods at the end of the paper. Let me now discuss the proof of Theorem.. The proof of Theorem. essentially splits into two parts. First, I improve on my earlier results from [2]. The improvement is that the initial conditions are no longer required to hold with large probability, but can hold only with small probability. Then the initial condition is verified by approximating the skew-shift model by the Almost Mathieu operator with potential 2λ cos(2π(βn + ω)). It is essential here, that one notices that if α is small one approximately has for m in a bounded range that α(n 0 + m) 2 αn αn 0 m. On a technical side except improving the lower bounds on the Lyapunov exponent, a second modification to the method of Bourgain in [] is that we do not perturb the Green s function directly, but work directly by perturbing the spectrum of the restricted operators. This enables one to prove the following extension Remark.3. Theorem. remains valid, if one replaces V α,λ (n) from (.) by (.5) V α,λ (n) = 2λ cos(2παn 2 ) + λκv ω (n) where V ω (n) is an ergodic potential and κ > 0 is small enough. The organisation of the paper is as follows. In Section 2, we discuss general results that imply positive Lyapunov exponent for ergodic Schrödinger operators. In Section 3, we apply these to the skew-shift potential and prove Theorem.. In Section 4, we prove the main result from Section 2. Finally, in Section 5 we discuss the modifications needed to proof Theorem Theorems that imply positive Lyapunov exponent In this section, I will discuss results that imply positive Lyapunov exponent. They all have in common, that they require assumptions with low probability. This is an essential difference to my earlier work [2], where I assumed that the bad events have small probability. In Section 7 and 8 of [2], Bourgain has stated

3 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL 3 similar results. However, he worked under Diophantine assumptions and used semi algebraic set techniques. We will work in the generality of ergodic Schrödinger operators. So let (Ω, µ) be a probability space, T : Ω Ω an invertible ergodic transformation, and f : Ω R a bounded measurable function. Given these, we introduce for ω Ω, the family of potentials (2.) V ω (n) = f(t n ω) for all integers n. We define the discrete Schrödinger operators H ω by (2.2) H ω : l 2 (Z) l 2 (Z) H ω u(n) = u(n + ) + u(n ) + V ω (n)u(n). These operators H ω will be bounded and self-adjoint operators. Define the n step transfer matrix n ( ) E Vω (n j) (2.3) A ω (E, n) =. 0 j=0 The Lyapunov exponent L(E) is given by (2.4) L(E) = lim log A ω (E, n) dµ(ω), n n Ω where the limit exists because of subadditivity. For a subinterval Λ Z, we introduce H ω,λ as the restriction of H ω to l 2 (Λ). Furthermore, we denote by σ(h) the spectrum of the operator H. The first version of our result is Theorem 2.. Given σ > 0, N, 0 < ε < 000, and E 0 R. Assume that (2.5) µ({ω : σ(h ω,[ N,N] ) [E 0 2ε, E 0 + 2ε] = }) σ. Furthermore, assume ε log(σ ) 2 75 and (2.6) N 50 ε 2. Then, there is a set E b [E 0 ε, E 0 + ε] of measure E b 2εe 2 75 ε for E [E 0 ε, E 0 + ε] \ E b (2.7) L(E) 250 σε. such that We will now reformulate this theorem in terms of the Green s function, since this language is more convenient for the proof. Denote by {e x } x Z the standard basis of l 2 (Z), that is {, x = n; (2.8) e x (n) = 0, otherwise. Given an interval Λ Z, E R, and x, y Λ, we introduce the Green s function (2.9) G ω,λ (E, x, y) = e x, (H ω,λ E) e y.

4 4 H. KRÜGER Definition 2.2. Given δ > 0, E R an interval, and N. We say that H ω is (δ, N, E)-good if there exists k [a, b] [ N, N] such that (2.0) sup Now, we come to sup E E x {a,b} G ω,[a,b] (E, k, x) e 2δ. Theorem 2.3. Let δ 4, E R an interval, and σ > 0. Assume that (2.) µ({ω : H ω is (δ, N, E) good}) σ. Furthermore assume the inequalities (2.2) δ 00 log( E ) e 2/75δ 226 e 3 (2.3) σ. Then there is a set E 0 E such that (2.4) E 0 ( e 75 δ ) E and for E E 0 (2.5) L(E) e σδ 2N. We now end this section, with that this theorem implies Theorem 2.. Proof of Theorem 2.. This follows from the Combes Thomas estimate (see e.g. Lemma 0. in [2]) with the choice δ = 2/ε. Remark 2.4. By changing the constants in Theorem 2., one could improve (2.6) to (2.6) N C log(ε ) ε for a large enough constant C > 0. I have decided not to do so, to keep the formulas simple. 3. Applications to the skew-shift We will begin by recasting the problem concerning the potential V (n) = 2λ cos(2παn 2 ) from (.) into the language of ergodic Schrödinger operators used in Section 2. Denote by T = R/Z the unit circle. As probability space we will use the torus T 2 equipped with the Lebesgue measure. For an irrational number α introduce the skew-shift T α by (3.) T α : T 2 T 2 T α (x, y) = (x + α, x + y) (mod ). It is known that T α is an uniquely ergodic and minimal map. Iterating the map T α on (x, y), we find (3.2) Tα n n(n ) (x, y) = (x + nα, y + nx + α) (mod ). 2 Write p 2 for the second coordinate projection p 2 (x, y) = y. We see that (3.3) p 2 (T n α ( α 2, 0)) = α 2 n2.

5 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL 5 Hence, it makes sense to introduce the skew-shift operator H α,x,y,λ by (3.4) H α,x,y,λ : l 2 (Z) l 2 (Z) H α,x,y,λ u(n) = u(n + ) + u(n ) + 2λ cos ( 2π p 2 ( T n α (x, y) )) u(n). We see that the operator H α,λ from the introduction will just be H 2α,α,0,λ. By (3.2), we approximately have for n in a bounded range and α small enough that p 2 (T n α (x, y)) y + nx. We will make this precise in Lemma 3.5. However, let us for now say, that we will try to approximate the skew-shift by rotations, and introduce these. For an irrational number α denote by R α : T T the rotation by (3.5) R α (x) = x + α (mod ). Introduce for α, ω, and λ > 0 the operator H R α,ω,λ by (3.6) H R α,ω,λ : l 2 (Z) l 2 (Z) H R α,ω,λu(n) = u(n + ) + u(n ) + 2λ cos(2π(nα + ω))u(n). This family of operators is known as Almost Mathieu Operator. We will need the following result Theorem 3.. Let δ > 0 and E satisfy (3.7) E [ 2 + δ, δ] [δ, 2 δ]. There are constants κ = κ(δ) > 0 and λ 0 = λ 0 (δ) > 0 such that for 0 < λ < λ 0, there is a set A [0, ] satisfying: (i) A κλ (ii) For α A (3.8) σ(h α ) [E λκ, E + λκ] =. Proof. This can be shown by a combination of Aubry Duality [9] as see in [] and the results of [3] or [0] on the spectrum of Hα,ω,λ R for large λ > 0. We will need the following lemma to pass from whole line operators to finite restrictions. Because of its abstract nature, we state it in the language of ergodic Schrödinger operators used in the previous section. Lemma 3.2. Let ε > 0, κ > 0, E 0 R, Q, Λ Z an interval, and Ω 0 Ω with µ(ω 0 ) > 0. Assume for ω Ω 0 that (3.9) σ(h ω ) [E 0 ε, E 0 + ε] =. Introduce for q = 0,..., Q the intervals (3.0) E q = [E 0 ε + 2ε q Q, E 0 ε + 2ε q + Q ]. There exists a set Q [0, Q ] satisfying (3.) #Q 8 κ such that for each q / Q, there is Ω = Ω (q) Ω 0 such that (3.2) µ(ω ) ( κ)µ(ω 0 )

6 6 H. KRÜGER and for ω Ω (3.3) σ(h ω,λ ) E q =. Proof. Let f(ω, q) be given by f(ω, q) = { σ(h ω,λ ) E q 0 otherwise. As usual, we let Λ c = Z \ Λ. We have H ω = H ω,λ H ω,λ c + K with K a rank 4 operator. Since E q E q consists of at most point for q q, we obtain that f(ω, q) 8. Hence, also Q q=0 Q q=0 f(ω, q)dµ(ω) 8. µ(ω 0 ) Ω 0 Define Q = {q : µ(ω 0) Ω 0 f(ω, q)dµ(ω) κ}. By Markov s inequality, we obtain (3.). This finishes the proof. In order to apply this lemma, we will need to subdivide [ 2 + δ, δ] [δ, 2 δ] into small intervals. To simplify the notation, we will only work with [δ, 2 δ]. First introduce for λ > 0 the subdivision (3.4) E 0 q (λ) = [δ + qκλ, δ + (q + )κλ], where q = 0,..., Q 0 (λ) with Q 0 (λ) = 2 δ κλ. Denote by τ > 0 the number from Theorem.. We will furthermore subdivide each of the Eq 0 (λ) = [E 0, E ] intervals into P = 256 τ pieces. These are given by [ (3.5) Eq,p(λ) = E 0 + p P (E E 0 ), E + p + 2 ] P (E E 0 ). We note that the intervals Eq,p(λ) overlap. However, if we only consider odd p respectively even p, we get non overlapping sets (except for the boundaries). We furthermore introduce [ (3.6) Eq,p(λ) 2 = E 0 + p P (E E 0 ), E + p + ] P (E E 0 ). We have that Lemma 3.3. We have that (3.7) Eq,p(λ) = 3κλ P, and Eq,p(λ) 2 Eq,p(λ). E 2 q,p(λ) = κλ P Proof. We observe that Eq 0 (λ) = κλ by construction, and that we divide into pieces of length 3 P E q 0 (λ) and P E q 0 (λ) respectively. Now, we come to Corollary 3.4. Let N and 0 q Q 0 (λ). For λ > 0 small enough, there exists a set P = P(q, λ) [0, P ] satisfying #P 64. Let p / P. Then there is Ω (λ) [0, ] 2 satisfying (3.8) Ω (λ) κλ 4

7 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL 7 and for (α, ω) Ω (λ), we have (3.9) σ(h R α,ω,λ,[ N,N] ) E q,p(λ) =. Proof. Let A be the set from Theorem 3. for E = δ + (2q + )κλ. Ω 0 = A [0, ] [0, ] 2. Then for (α, ω) Ω 0, we have σ(h R α,ω,λ) E 0 q (λ) = Introduce by Theorem 3.. Now apply Lemma 3.2 with Q = P, κ = 4, and Λ = [ N, N] to the set of Eq,p with odd respectively even p. The claim follows. The next lemma follows from (3.2). Lemma 3.5. Assume that α 2ε N 2, then (3.20) p 2 (T n α (x, y)) R y (x) ε for n N. We have the following lemma, which allows us to compare the operators H R x,y,λ,[ N,N] and H α,x,y,λ,[ N,N] for α sufficiently small. Lemma 3.6. Let ε > 0 and N. Then for α 2ε N 2, δ > 0. and E R such that (3.2) σ(hx,y,λ,[ N,N] R ) [E δ, E + δ] =. We also have that (3.22) σ(h α,x,y,λ,[ N,N] ) [E (δ λε), E + (δ λε)] =. Proof. This follows from the last lemma, and that the spectrum we have for the spectrum dist(σ(h), σ(h + V )) V, where dist denotes the Hausdorff metric. Combining this lemma with Corollary 3.4, we obtain Corollary 3.7. Assume (3.23) α κ 2 P N 2. Given q, denote by P the set from Corollary 3.4. For p / P, there is a set Ω of (x, y) such that (3.24) Ω κλ 4 and for (x, y) Ω, we have that (3.25) σ(h α,x,y,λ,[ N,N] ) Eq,p(λ) 2 =. Proof. We first note that Eq,p(λ) is the κλ P neighborhood of E q,p(λ). 2 Hence, we will apply Lemma 3.6 with ε = κλ P to the sets from Corollary 3.4 to obtain the result. We now come to

8 8 H. KRÜGER Proof of Theorem.. We just prove the claim for [δ, 2 δ], the interval [ 2+δ, δ] can be dealt with similarly. First obverse by construction that [δ, 2 δ] \ Eq,p(λ) 2 τ 4. q,p/ P(q) Furthermore by the previous corollary, we have that we may apply Theorem 2. with the choices σ = κλ 4, ε = κλ 2P, N = 50 ε as long as λ is sufficiently small, and α 2 satisfies the conditions from the previous corollary. We see that by choosing again λ > 0 sufficiently small, we can ensure that E b τ 4 QP, since QP λ. An inspection of the above shows, that we prove the bound L α,λ (E) κ2 Since P 300 τ, we conclude that 500 P λ2. L α,λ (E) κ2 τ λ2, which justifies the discussion following the statement of Theorem.. The current proof shows that we have α 0 λ in Theorem.. Taking Re- 4 mark 2.4 into account this can be improved to α 0 log(λ)2 λ Proof of the Theorem 2.3 We now proceed to prove Theorem 2.3. The proof will follow by an adaptation of the arguments from my earlier work [2]. The main difference is that we will have to deal with small probabilities. We will need a variant of Theorem 3.3. from [2], which can be extracted from the proof. Theorem 4.. Let K, δ 4, and E R an interval. Assume that for a positive measure set of ω, we can find sequences N t = N t (ω), L t = L t (ω) such that N t (4.) lim K t L t and for t there are integers k l = k l (t, ω) (4.2) 0 k 0 < k < k 2 < < k L < k Lt+ N t and a set L t = L t (ω) [, L t ] such that for l / L t, we have that (4.3) G ω,[kl +,k l+ ](E, k l, k l± ) 2 e δ for E E. Furthermore assume the inequalities (4.4) (4.5) Then for a set E 0 satisfying δ 00 log( E ) e 2 75 δ 2 25 e 3 K 3 (4.6) E 0 ( e 2 25 δ ) E,

9 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL 9 we have for E E 0 (4.7) L(E) e Proof. This result can be seen by verifying that our assumptions are the same as the conclusions of Lemma 9., after which one can proceed as in the proof of Theorem 3.3. in [2]. We furthermore specialize to the case σ 4. δ K. For the proof of Theorem.2, we will furthermore need Corollary 4.2. Denote by E 0 the set from the previous theorem. Let E E 0. Denote by Ω 0 the set of ω, where the assumptions of the previous theorem hold for fixed large enough t. Let N = Nt 2. For ω Ω 0, we have that (4.8) (4.9) (4.0) N log A ω(e, N) e 99 δ 99 K N log A T N ω(e, N) e 99 δ 99 K 2N log A ω(e, 2N) e 99 δ 99 K. Proof. This follows by the methods used to prove the previous theorem and the results in Section 5 of my earlier paper [2], in particular Lemma 5.2. We will furthermore need Lemma 4.. from [2]. Lemma 4.3. Let T : Ω Ω be an ergodic transformation, Ω g Ω, K. Then there exists Ω 0 Ω such that for ω Ω 0 there is a sequence L t = L t (ω) such that (4.) #{0 l L t : T lk ω Ω g } µ(ω g) L t 2 and µ(ω 0 ) > 0. We apply this lemma to set from (2.). By this, we obtain Lemma 4.4. Assume (2.). For ω from a set of positive measure, we can find sequences L t, N t such that (i) We have (4.2) L t N t σ 2. (ii) There are integers a l = a l (t), b l = b l (t), and k l = k l (t) such that (4.3) 0 < a < k < b < a 2 < k 2 < b 2 < k Lt < b Lt < N t. Here the k l can depend on t. (iii) For each l, we have that (4.4) sup G ω,[al,b l ](E, k l, x) e 2δ. x {a l,b l } Proof. Apply the previous lemma to the set from (2.) with K = 2N +. For the proof of Theorem.2, we remark

10 0 H. KRÜGER Remark 4.5. By replacing the use of Lemma 4.. in [2] by Lemma 4.2. we can even ensure that the result of this lemma holds for a set of ω, whose measure is arbitarily close to. Hence, by an inspection of the following, one can conclude that the conclusions of Corollary 4.2 hold for a set Ω 0, whose measure is arbitarily close to. We now fix ω such that the conclusions of Lemma 4.4 hold. The main problem with (4.4) is that the intervals [a l, b l ] on which it holds are not overlapping. Hence, there is a priori no way one could iterate it to obtain an estimate for the Green s function on all intervals. So, we will now proceed to make this estimate hold on overlapping intervals as required by the results from my earlier work. The strategy of this will be similar to one in [2], we will obtain a priori bounds on the large intervals by being far from the spectrum, and then use this to improve our current bounds. For this, we first need. Lemma 4.6. There is L [, L t ] satisfying (4.5) #L L t 8 such that for l / L (4.6) k l+ k l 32 σ. Proof. We have that L t k l+ k l 2 N t l= Hence, the claim follows by Markov s inequality and (4.2). In order to satisfy (4.4), we will subdivide E = [E 0, E ] into intervals E q. So define Q = 4 (E E 0 )e δ and (4.7) E q = [E 0 + q E E 0 Q, E + (q + ) E E 0 Q ], for q = 0,..., Q. We then have that (4.8) E q 4e δ. We furthermore have that 4e δ e δ/00 since by assumption δ. We will now need to eliminate eigenvalues. This will be accomplished by the following lemma. Lemma 4.7. There exists a set Q satisfying (4.9) #Q 800 σ and for q / Q, there exists L 2 = L 2 (q) such that (4.20) #L 2 L t 8 and for l / L L 2 (4.2) dist(e q, σ(h ω,[kl +,k l+ ])) > 4e δ.

11 ON POSITIVE LYAPUNOV EXPONENT FOR THE SKEW-SHIFT POTENTIAL Proof. We follow the proof of Lemma 6.6. in [2]. For each l introduce g(l) = #{q : dist(e q, σ(h ω,[kl +,k l+ ])) 4e δ }. By the previous lemma, we have for l / L that g(l) 96/σ 00/σ. Hence, q #{l / L : dist(e q, σ(h ω,[kl +,k L l+ ])) 4e δ } 00 t σ. The claim now follows from Markov s inequality. We observe that the measure of the set of energies, we need to eliminate can be estimate by (4.22) Lemma 4.8. If σ E e δ. (4.23) dist(e, σ(h ω,[kl +,k l+ ])) 4e δ. holds, then (4.24) G ω,[kl +,k l+ ](E, k l, k l± ) 2 e δ. Proof. By the resolvent equation, we have that G ω,[kl +,k l+ ](E, k l, k l± ) = Hence, we obtain that G ω,[kl +,k l+ ](E, a l, k l± )G ω,[al,b l ](E, k l, a l ) G ω,[kl +,k l+ ](E, b l +, k l± )G ω,[al,b l ](E, k l, b l ). G ω,[kl +,k l+ ](E, k l, k l± ) 2 ( 4 eδ ) e 2δ. This finishes the proof. Proof of Theorem 2.3. This follows from Theorem 4., since it is applicable by the previous lemmas. 5. Proof of Theorem.2 Due to Corollary 4.2 and Remark 4.5, we are able to satisfy the conditions of Lemma 2.8. from Bourgain, Goldstein, and Schlag [6]. Hence, we might iterate it as described in Subsection 2.5. of [6]. This allows us to conclude Theorem.2. It should be remarked here, that we are not able to improve Theorem 2.3, since the methods of [6] require the particular situation of the skew-shift with an analytic sampling function. Already the extension to higher skew-shifts as considered in Theorem 2.7. of [2] seems non trivial. I believe that following the approach of [5] as explained in [6], one should also be able to conclude Anderson localization outside the set E b. Unfortunately, we cannot ensure that there is spectrum there.

12 2 H. KRÜGER References [] J. Bourgain, Positive Lyapounov exponents for most energies, Geometric aspects of functional analysis, 37 66, Lecture Notes in Math. 745, Springer, Berlin, [2] J. Bourgain, On the spectrum of lattice Schrödinger operators with deterministic potential, J. Anal. Math. 87 (2002), [3] J. Bourgain, Estimates on Green s functions, localization and the quantum kicked rotor model, Ann. of Math. (2) 56- (2002), [4] J. Bourgain, Green s function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, 58. Princeton University Press, Princeton, NJ, x+73 pp. [5] J. Bourgain, M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. Math., 52 (2000), [6] J. Bourgain, M. Golstein, W. Schlag, Anderson localization for Schrödinger operators on Z with potentials given by the skew-shift, Comm. Math. Phys (200), [7] D. Damanik, Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon s 60th birthday, , Proc. Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007 [8] M. Disertori, W. Kirsch, A. Klein, F. Klopp, V. Rivasseau, Random Schrödinger Operators, Panoramas et Synthses 25 (2008), xiv + 23 pages [9] A. Y. Gordon, S. Jitomirskaya, Y. Last, B. Simon, Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math. 78:s (997), [0] B. Helffer, P. Kerdelhué, J. Sjöstrand, Le papillon de Hofstadter revisité. [Hofstadter s butterfly revisited] Mém. Soc. Math. France, 43 (990), 87 pp. [] M. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d un théorème d Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv 58 (983), [2] H. Krüger, Multiscale analysis for Ergodic Schr dinger operators and positivity of Lyapunov exponents, preprint. [3] H. Krüger, Semiclassical analysis of the largest gap of quasi-periodic Schrödinger operators, preprint. [4] W. Schlag, On discrete Schrödinger operators with stochastic potentials, XIVth International Congress on Mathematical Physics, , World Sci. Publ., Hackensack, NJ, [5] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, Department of Mathematics, Rice University, Houston, TX 77005, USA address: helge.krueger@rice.edu URL: