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Universal hierarchical structure of quasiperiodic eigenfunctions

We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.


Introduction
A very captivating question and a longstanding theoretical challenge in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. Such structure was first predicted in the work of Azbel in 1964 [11]. It was numerically confirmed through the famous butterfly plot and further argued for by Hofstadter in [29], for the spectrum of the almost Mathieu operator. This was even before this model was linked to the integer quantum Hall effect [48] and other important phenomena. Mathematically, it is known that the spectrum is a Cantor set for all irrational fluxes [5], and moreover, even all gaps predicted by the gap labeling are open in the non-critical case [6,8]. Both were very important challenges in themselves, even though these results, while strongly indicate, do not describe or explain the hierarchical structure, and the problem of its description/explanation remains open, even in physics. As for the understanding the hierarchical behavior of the eigenfunctions, despite significant numerical studies and even a discovery of Bethe Ansatz solutions [1,49] it has also remained an important open challenge even at the physics level. Certain results indicating the hierarchical structure in the corresponding semi-classical/perturbative regimes were previously obtained in the works of Sinai, Helffer-Sjostrand, and Buslaev-Fedotov (see [22,28,47], and also [52] for a different model ).
In this paper we address the latter problem by describing the universal self-similar exponential structure of eigenfunctions throughout the entire localization regime. We determine explicit universal functions f (k) and g(k), depending only on the Lyapunov exponent and the position of k in the hierarchy defined by the denominators q n of the continued fraction approximants of the flux α, that completely define the exponential behavior of, correspondingly, eigenfunctions and norms of the transfer matrices of the almost Mathieu operators, for all eigenvalues corresponding to a.e. phase , see Theorem 2.1. 1 Our result holds for all frequency and coupling pairs in the localization regime. Since the behavior is fully determined by the frequency and does not depend on the phase, it is the same, eventually, around any starting point, so is also seen unfolding at different scales when magnified around local eigenfunction maxima, thus describing the exponential universality in the hierarchical structure, see, for example, Theorems 2.2,2.4.
While the almost Mathieu family is precisely the one of main interest in physics literature, it also presents the simplest case of analytic quasiperiodic operator, so a natural question is which features discovered for the almost Mathieu would hold for this more general class. Not all do, in particular, the ones that exploit the self-dual nature of the family H λ,α,θ often cannot be expected to hold in general. In case of Theorems 2.1 and 2.2, we conjecture that they should in fact hold for general analytic (or even more general) potentials, for a.e. phase and with ln |λ| replaced by the Lyapunov exponent L(E) (see Footnote 4), but with otherwise the same or very similar statements. The hierarchical structure theorems 2.2 and 2.4 are also expected to hold universally for most (albeit not all, as in the present paper) appropriate local maxima. Some of our qualitative corollaries may hold in even higher generality. Establishing this fully would require certain new ideas since so far even an arithmetic version of localization for the Diophantine case has not been established for the general analytic family, the current state-ofthe-art result by Bourgain-Goldstein [18] being measure theoretic in α. However, some ideas of our method can already be transferred to general trigonometric polynomials [35]. Moreover, our method was used recently in [27] to show that the same f and g govern the asymptotics of eigenfunctions and universality around the local maxima throughout the a.e. localization regime in another popular object, the Maryland model.
Since we are interested in exponential growth/decay, the behavior of f and g becomes most interesting in case of frequencies with exponential rate of approximation by the rationals. In general, localization for quasiperiodic operators is a classical case of a small denominator problem, and has been traditionally approached in a perturbative way: through KAM-type schemes for large couplings [21,24,47] (which, being KAM-type schemes, all required Diophantine conditions on frequencies) or through perturbation of periodic operators (Liouville frequency). Unlike the random case, where, in dimension one, localization holds for all couplings, a distinctive feature of quasiperiodic operators is the presence of metal-insulator transitions as couplings increase. Even when non-perturbative methods, for the almost Mathieu and then for general analytic potentials, were developed in the 90s [18,31], allowing to obtain localization for a.e. frequency throughout the regime of positive Lyapunov exponents, they still required Diophantine conditions, and exponentially approximated frequencies that are neither far from nor close enough to rationals remained a challenge, as for them there was nothing left to perturb about or to remove. Moreover, it has become clear that for all frequencies, the true localization threshold should be arithmetically determined and happen precisely where the exponential growth provided by the Lyapunov exponent beats the exponential strength of the small denominators. Thus the most interesting regime -the neighborhood of the transition -required dealing with the exponential frequencies not amenable to perturbations/parameter removals, adding a strong number theoretic flavor to the problem. The precise second transition conjecture was stated for the almost Mathieu operator [30]. Our analysis provides also a (constructive) solution to the full arithmetic version of the transition in frequency and explains the role of frequency resonances in the phenomenon of localization, in a sharp way.
It is the central quasiperiodic model due to coming from physics and attracting continued interest there. First appearing in Peierls [44], it arises as related, in two different ways, to a two-dimensional electron subject to a perpendicular magnetic field and plays a central role in the Thouless et al theory of the integer quantum Hall effect. For further background, history, and surveys of results see [20,32,38,40] and references therein.
Almost Mathieu operator has a transition from zero to positive Lyapunov exponents on the spectrum at |λ| = 1 (the critical coupling) leading to the conjecture, dating back to [2], that it induces a transition from absolutely continuous to pure point spectrum. For Diophantine α this was proved in [31]. The result was extended to all α, θ for |λ| < 1 (the subcritical regime) in [4], solving one of the Simon's problems [46]. For the supercritical regime (|λ| > 1) it is known however that the nature of the spectrum should depend on the arithmetic properties of α [10,26]. Set where pn qn are the continued fraction approximants of α. For any irrational number α, we say that phase θ ∈ (0, 1) is Diophantine with respect to α, if there exist κ > 0 and ν > 0 such that for any k ∈ Z\{0}, where ||x|| R/Z = dist(x, Z). Clearly, for any irrational number α, the set of phases which are Diophantine with respect to α is of full Lebesgue measure. The conjecture in [30] states that for α-Diophantine (thus almost every) θ, H λ,α,θ satisfies Anderson localization (i.e., has only pure point spectrum with exponentially decaying eigenfunctions) if |λ| > e β , and has, for all θ, purely singular continuous spectrum for 1 < |λ| < e β . 2 For β = 0 this follows from [31]. A progress towards the localization side of the above conjecture was made in [5] (localization for |λ| > e us to find precise asymptotics of the norms of the transfer-matrices, providing the first example of this sort for non-uniformly hyperbolic dynamics. Since those norms sometimes differ significantly from the reciprocals of the eigenfunctions, this leads to further interesting and unusual consequencies, for example exponential tangencies between contracted and expanded directions at the resonant sites. From this point of view, our analysis also provides, as far as we know, the first study of the dynamics of Lyapunov-Perron non-regular points, in a natural setting. An artificial example of irregular dynamics can be found in [12], p.23, however it is not even a cocycle over an ergodic transformation, and we are not aware of other such, even artificial, ergodic examples where the dynamics has been studied. Loosely, for a cocycle A over a transformation f acting on a space X (Lyapunov-Perron) non-regular points x ∈ X are the ones at which Oseledets multiplicative ergodic theorem does not hold coherently in both directions. They therefore form a measure zero set with respect to any invariant measure on X. 3 Yet, it is precisely the non-regular points that are of interest in the study of Schrödinger cocycles in the non-uniformly hyperbolic (positive Lyapunov exponent) regime, since spectral measures, for every fixed phase, are always supported on energies where there exists a solution polynomially bounded in both directions, so the (hyperbolic) cocycle defined at such energies is always nonregular at precisely the relevant phases. Thus the non-regular points capture the entire action from the point of view of spectral theory, so become the most important ones to study. One can also discuss stronger non-regularity notions: absence of forward regularity and, even stronger, non-exactness of the Lyapunov exponent [12]. While it is not difficult to see that energies in the support of singular continuous spectral measure in the non-uniformly hyperbolic regime always provide examples of non-exactness, our analysis gives the first non-trivial example of non-exactness with non-zero upper limit (Corollary 2.13). Finally, as we understand, this work provides also the first natural example of an even stronger manifestation of the lack of regularity, the exponential tangencies (Corollary 2.14). Tangencies between contracted and expanded directions are a characteristic feature of nonuniform hyperbolicity (and, in particular, always happen at the maxima of the eigenfunctions). They complicate proofs of positivity of the Lyapunov exponents and are viewed as a difficulty to avoid through e.g. the parameter exclusion [15,17,51]. However, when the tangencies are only subexponentially deep they do not in themselves lead to non-exactness. Here we observe the first natural example of exponentially strong tangencies (with the rate determined by the arithmetics of α and the positions precisely along the sequence of resonances.) The localization-for-the-exponential-regime method of [5] consists of different arguments for non-resonant (meaning sufficiently far from jq n on the corresponding scale) sites and for the resonant ones (the rest). It is the resonant sites that lead to dealing with the smallest denominators and that necessitate the |λ| > e 16 9 β requirement in [5]. Here we start with the same basic setup, and only technically modify the non-resonant statement of [5]. However we develop a completely new bootstrap technique to handle the resonant sites, allowing us to get to the transition and obtain the fine estimates. The estimates from below (that coincide with our estimates from above) are also new. In general, the statements that are technically similar to the ones in the existing literature are collected in the Appendices, while all the results/proofs in the body of the paper are, in their pivotal parts, not like anything that has appeared before.
The key elements of the technique developed in this paper are robust and have made it possible to approach other scenarios. As such, in the upcoming work we prove the sharp phase transition for Diophantine α and all θ and establish sharp exponential asymptotics of eigenfunctions and transfer matrices in the corresponding pure point regime [36]. Moreover, our analysis reveals there a universal reflection-hierarchical structure in the entire regime of phaseinduced resonances, a phenomenon not even previously discovered in physics literature. Thus while in this paper we develop a complete understanding of frequency induced resonances, in [36] we further develop some of the key ideas of this manuscript to obtain a complete understanding of phase induced resonances. In other follow-up works we determine the exact exponent of the exponential decay rate in expectation for the Diophantine case [34] and study delicate properties of the singular continuous regime, obtaining upper bounds on fractal dimensions of the spectral measure and quantum dynamics for the almost Mathieu operator [37], as well as potentials defined by general trigonometric polynomials [35].
Except for a few standard, general (e.g. uniform upper semicontinuity) or very simple to verify statements, this paper is entirely self contained. The only technically involved fact that we use without proving it in the paper is Lemma 3.1 [19] but this is not even necessary if we replace ln |λ| by the Lyapunov exponent L(E) throughout the manuscript. 4
Set alsor n 0 = 1 for convenience. Notice that f, g only depend on α. f (k) decays and g(k) grows exponentially, at varying rates that depend on the position of k in the hierarchy defined by the continued fraction expansion of α, see Fig. 1.
Our first main result is that in the entire regime |λ| > e β , the exponential asymptotics of the generalized eigenfunctions and norms of transfer matrices at the generalized eigenvalues are completely determined by f (k), g(k).
Theorem 2.1. Let α ∈ R\Q be such that |λ| > e β(α) . Suppose θ is Diophantine with respect to α, E is a generalized eigenvalue of H λ,α,θ and φ is the generalized eigenfunction. Let . Then for any ε > 0, there exists K (depending on λ, α,Ĉ, ε and Diophantine constants κ, τ ) such that for any |k| ≥ K, U (k) and A k satisfy ℓq n (ℓ + 1)q n (ℓ + 2)q n (ℓ + 3)q n (ℓ + 4)q n k qn+1 2 Certainly, there is nothing special about k = 0, so the behavior described in Theorem 2.1 happens around arbitrary point k = k 0 . This implies the self-similar nature of the eigenfunctions): U (k) behave as described at scale q n but when looked at in windows of size q k , q k < q n−1 will demonstrate the same universal behavior around appropriate local maxima/minima.

.
If k 0 is strongly nonresonant, then for all 2s ∈ I j ς1,ς2 . Remark 2.3. Actually, a modification in our proof allows to formulate (16) in Theorem 2.2 with non-resonant condition (14) only required for 2q j−1 < |k| ≤ q j .
In case β(α) > 0, Theorem 2.1 also guarantees an abundance (and a hierarchical structure) of local maxima of each eigenfunction.
Moreover, every local n j−s -maximum on the interval b an j ,an j−1 ,...,an j−s+1 +[−e ǫ ln λqn j−s , e ǫ ln λqn j−s ] is of the form b an j ,an j−1 ,...,an j−s for some a nj−s .
Remark 2.5. By I of Theorem 2.4, the local maximum can be determined up to a constant K 0 = qn 0+1 . Actually, if k 0 is only a local n j + 1-maximum, we can still make sure that I, II and III of Theorem 2.4 hold. This is the local version of Theorem 2.4, see Theorem 7.3.
Remark 2.6. qn 0 +1 is the scale at which phase resonances of θ+k 0 α still can appear. Notably, it determines the precision of pinpointing local n j -maxima in a (exponentially large in q nj ) neighborhood of k 0 , for any j. When we go down the hierarchy, the precision decreases, but note that except for the very last scale it stays at least iterated logarithmically 6 small in the corresponding scale q nj−s Thus for x ∈ b an j ,an j−1 ,...,an j−s + [− cj−s 2 q nj−s , 1 2 q nj−s ], the behavior of φ(x) is described by the same universal f in each q nj−s -window around the corresponding local maximum b an j ,an j−1 ,...,an j−s ,s = 0, 1, ..., k. We call such a structure hierarchical, and we will say that a local j-maximum is k-hierarchical if the complete hierarchy goes down at least k levels (for a precise definition see Section 7). We then have an immediate corollary 5 If there are several, what follows is true for each. 6 for most scale even much less Corollary 2.7. There exists C = C(α, λ, κ, ν, ǫ) such that every local n j -maximum in [k 0 − e ς ln |λ|qn j , k 0 + e ς ln |λ|qn j ] is at least (j/2 − C)-hierarchical.
Remark 2.8. The estimate on the depth of the hierarchy in the corollary assumes the worst case scenario when all scales aftern 0 are Liouville. Otherwise the hierarchical structure will go even much deeper. Note that a local n j -maximum that is not an n j+1 -maximum cannot be k-hierarchical for k > j.
Universal hierarchical structure of an eigenfunction Local maximum Local maximum
This solves the arithmetic version of the second transition conjecture in that it establishes localization throughout the entire regime of (α, λ) where localization may hold for any θ (see the discussion in the introduction), for an arithmetically defined full measure set of θ.
We note that Theorem 2.9 cannot be upgraded to all θ in the regime |λ| > e β [39] so exclusion of a certain arithmetically defined set where the spectrum must be singular continuous is necessary. There is a conjecture of where in this regime the transition in θ happens [30] but we do not explore it in this work. The sharp transition in θ for Diophantine α will be established in the follow-up work [36]. Also, it could be added that, for all θ, H λ,α,θ has no localization (i.e., no exponentially decaying eigenfunctions) if |λ| = e β (see Appendix A.1).
Remark 2.10. Theorems 2.1, 2.9 cover the optimal range of (α, λ) for a.e. θ. For Theorem 2.9, even though some θ have to be excluded [39], we do not claim the Diophantine condition on θ is optimal. At the same time, exponentially strong θ-resonances (exponentially small lower bound in (2) instead of a polynomial) will make Theorem 2.1 false as stated, no matter how small the exponent, and would require differently defined f and g. In [36] we obtain f ′ and g ′ that govern the exponential behavior of eigenfunctions and transfermatrices for all θ throughout the entire pure point regime corresponding to Diophantine α.
Let ψ(k) denote any solution to H λ,α,θ ψ = Eψ that is linearly independent with respect to . An immediate counterpart of (13) is the following Corollary 2.11. Under the conditions of Theorem 2.1 for large k vectorsŨ (k) satisfy Thus every solution is expanding at the rate g(k) except for one that is exponentially decaying at the rate f (k).
It is well known that for E in the spectrum the dynamics of the transfer-matrix cocycle A k is nonuniformly hyperbolic. Moreover, E being a generalized eigenvalue of H λ,α,θ already implies that the behavior of A k is non-regular. Theorem 2.1 provides precise information on how the non-regular behavior unfolds in this case. Previously, a study of some features of the non-regular behavior for the almost Mathieu operator was made in [23]. We are not aware though of other non-artificially constructed examples of non-uniformly hyperbolic systems where non-regular behavior can be described with such precision as in the present work.
The information provided by Theorem 2.1 leads to many interesting corollaries which will be explored elsewhere. Here we only want to list a few immediate sharp consequences.
iii) Outside an explicit sequence of lower density zero, 7 Therefore the Lyapunov behavior for the norm fails to hold only along a sequence of density zero. It is interesting that the situation is different for the eigenfunctions. While, just like the overall growth of A k is ln |λ| − β, the overall rate of decay of the eigenfunctions is also ln |λ| − β, they however decay at the Lyapunov rate only outside a sequence of positive upper density. That is ii) iii) There is an explicit sequence of upper density iv) There is an explicit sequence of upper density 1 2 β ln |λ| , 9 along which The fact that g is not always the reciprocal of f leads also to another interesting phenomenon.
Let 0 ≤ δ k ≤ π 2 be the angle between vectors U (k) andŨ (k). Corollary 2.14. We have As becomes clear from the proof, neighborhoods of resonances q n are the places of exponential tangencies between contracted and expanded directions, with the rate approaching −β along a subsequence. 10 Exponential tangencies also happen around points of the form jq n but at lower strength. This means, in particular, that A k with k ∼ q n is exponentially close to a matrix with the trace e (ln |λ|−β)k .
The rest of this paper is organized in the following way. We list the definitions and standard preliminaries in Section 3. We also include there the non-resonant regularity statement. While similar to the corresponding statements in [5,42,43], it differs in enough technical details that a proof is needed for completeness. We present this proof in Appendix B. Section 4 is devoted to the bootstrap localization argument, establishing sharp upper bounds for the resonant case. Section 5 is devoted to the lower bounds. In Section 6 we prove the statements about eigenfunctions: (12) of Theorem 2.1, Theorems 2.2 and 2.9. In Section 7, we will prove the hierarchical structure Theorem 2.4 and Corollary 2.7. In Section 8, we study the growth of transfer matrices and prove (13) of Theorem 2.1. The remaining Corollaries are proved in Section 9.

Preliminaries
Fix α ∈ R\Q such that β(α) < ∞. Unless stated otherwise, we always assume λ > e β (for λ < −e β , notice that H λ,α,θ = H −λ,α,θ+ 1 2 ), θ is Diophantine with respect to α and E is a generalized eigenvalue. We also assume φ is the corresponding generalized eigenfunction of 8 It will be clear from the proof that the sequence contains ⌊ qn 2 ⌋, n = 1, · · · . 9 As will be clear from the proof, this sequence can have lower density ranging from 0 to 1 2 β ln |λ| depending on finer continued fraction properties of α. 10 In fact the rate is close to − ln q n+1 qn for any large n.
The Lyapunov exponent is given by The Lyapunov exponent can be computed precisely for E in the spectrum of H λ,α,θ . We denote the spectrum by Σ λ,α (it does not depend on θ).
We start with the basic setup going back to [31]. Let us denote It is easy to check that .
It is easy to check that We have the following Lemma.
We say θ is n-Diophantine with respect to α, if for some κ > 0, ν > 1 the following hold for all |k| ≤ 2q n and (36) ||2θ for all 2q n < |k| ≤ 2q n+1 . Define b n = q t n with 8 9 ≤ t < 1 (t will be defined later). For any k > 0, we will distinguish two cases with respect to n: (i) |k − ℓq n | ≤ b n for some ℓ ≥ 1, called n−resonance.
For the n−nonresonant y, let n 0 be the least positive integer such that 4q n−n0 ≤ dist(y, q n Z). Let s be the largest positive integer such that 4sq n−n0 ≤ dist(y, q n Z). Notice that n 0 ≤ C(α).
The following theorem is similar to a statement appearing in [5] with modifications in [42,43]. We present a proof in Appendix B.
Remark 3.8. In the present paper, we only use Theorem 3.5 with C = 50C ⋆ , where C ⋆ is given by (37) (see the next section).

Bootstrap resonant localization
In this section we assume θ is n−Diophantine with respect to α. Clearly, it is enough to consider k > 0. In this section we study the resonant case. Suppose there exists some where ⌊m⌋ denotes the smallest integer not exceeding m.
For an arbitrary solution ϕ satisfying Hϕ = Eϕ, let qn . Fix ψ satisfying (23) and denote by R n j = r n,ψ j , and r n j = r n,φ j . Since we keep n fixed in this section we omit the dependence on n from the notation and write r ϕ j , R j , and r j . Note that below we always assume n is large enough. 11 In the next Lemma and its variant, Lemma 4.2, we establish exponential decay of the eigenfunctions at non-resonant points, at the nearly Lyapunov rate, with respect to the distance to the resonances.
qn and b n+1 ≥ qn 2 , or ii) j = 0, then for sufficiently large n, Proof. The proof builds on the ideas used in the proof of Lemma 9.11 in [5] and Lemma 3.2 in [42]. However it requires a more careful approach.
We first prove the case i).

Lower bounds on decaying solution in the resonant case
In this section we assume θ is n−Diophantine with respect to α. We will study the lower bound on φ for the resonant sites. Recall that b n+1 ≥ qn 2 in this case.
We first list two standard facts.
Using the fact |e y − 1| ≤ ye y for y > 0, we obtain Combining this with (73) completes the proof.
This is in contradiction with (80). Thus by (82), we have where the second inequality holds by (81) and the third inequality hold by (76).
By (72), the following holds where the third inequality holds by (84).
Proof of Theorem 5.1.
Proof. We can proceed by induction. Set j = 0 in Lemma 5.4. By (55), the second case (75) can not happen, thus Theorem 5.1 holds for j = 1.
We will show (70) holds for p = j. Let us apply Lemma 5.4 to p = j − 1. If (74) holds for p = j − 1, the result follows. Otherwise by (75), we havẽ where the second inequality holds by (86).
6. Decaying solutions. Proof of (12), Theorems 2.2, 2.4 and 2.9 In this section the dependence on n will play a role, so we go back to the r n j ,r n j notation. We first give a series of auxiliary facts. Recall footnote 11. Theorem 6.1. Assume θ is n−Diophantine with respect to α. For any 1 ≤ j ≤ 10 bn+1 qn with b n+1 ≥ qn 2 , we haver n j e −εjqn ≤ r n j ≤r n j e εjqn andr n j e −εjqn ≤r n j ≤r n j e εjqn . Proof. For any ε > 0, we choose η small enough.Using (55) j times, we have Similarly, using (70) j times, we havẽ r n j ≥ q j n+1 j! exp{−(ln λ + ε)jq n }.
By Stirling formula and (53), we obtain the theorem.
Case 2: k ≥ q t0 n−n0+1 . Then where the third inequality holds by the definition of t 0 . Noting that k ≤ q n−n0+1 , one has Similarly, exp{−(j + 1)q n−n0 (ln λ + ε)} ≤r n−n0 j+1 ≤ exp{−(j + 1)q n−n0 (ln λ − ε)}. Thus in order to prove case 2, it suffices to show The left inequality holds by (53). We start to prove the right inequality. For any y ∈ [εk, k] or y ∈ [q n − k, q n − εk], let n ′ 0 be the least positive integer such that 4q n−n ′ 0 ≤ dist(y, q n Z), thus n ′ 0 ≥ n 0 . Let s be the largest positive integer such that 4sq n−n ′ 0 ≤ dist(y, q n Z). Set I 1 , I 2 ⊂ Z as follows , and let θ j = θ + jα for j ∈ I 1 ∪ I 2 . The set {θ j } j∈I1∪I2 consists of 4sq n−n ′ 0 elements. By case ii) of Theorem 3.5, y is (ln λ + 8 ln(sq This implies for any y ∈ For any y ∈ (k, q n − k), let s be the largest positive integer such that sq n−n0 ≤ dist(y, q n Z) and set I 1 , I 2 ⊂ Z as follows: By the same reason, (94) and (95) also hold for n ′ 0 = n 0 . Arguing exactly as in the proof of Lemma 4.1, with (39) replaced with (94) and (40) with (95), we obtain this implies (93) and thus the theorem.
Remark 6.4. The assumption that θ is n − n 0 Diophantine with respect to α is sufficient for the proof of case 1. The assumption that θ satisfies (35) is sufficient for the proof of case ii) of Theorem 3.5, and therefore for the proof of case 2. Then by Remark 3.6, the assumption that θ is n − 1 Diophantine with respect to α is sufficient for the proof of case 2.
Remark 6.5. Suppose we only consider q 8 9 n ≤ k ≤ cq n with c ≤ 1 2 in Theorem 6.3. Theorem 6.3 still holds if we only have (11) for function φ(k) on [−cq n , 2cq n ] for some c > 0.
Case ii.2: q n ≤ k ≤ qn+1 2 In this case (97) follows directly from Theorem 6.3 (with n + 1 − n 0 = n), because n 0 = 1 so that the fact θ is n Diophantine can guarantee both cases 1 and 2 of Theorem 6.3.

Proof of Theorem 2.2
Proof. The proof follows that of (97) by shifting by k 0 units, and Remark 6.5.
Proof of Theorem 2.9.
Proof. Assume θ is Diophantine with respect to α. First by the definition of β(α), one has for any large n and any ℓr n ℓ ≤ e −(ln λ−β−ε)ℓqn . Combining with the definition of f (k) and (12), we have We therefore established that every generalized eigenfunction decays exponentially, which by Schnol's Theorem [16] implies the localization statement. By the definition of β(α) again, there exists a subsequence q n k of q n such that (99) q n k +1 ≥ e (β−ε)qn k .

Hierarchical structure
As we have already established Theorem 2.9 we know that each generalized eigenfunction decays exponentially so has a global maximum. Assume its global maximum (see Footnote 2) is at 0 and φ is normalized by φ ∞ = 1. Note that thenĈ in (11) is equal to 1 so all dependence of the largeness of n on E (see Footnote 12) disappears. Theorem 2.1 provides, for sufficiently large n plenty of local n-maxima in the vicinity of aq n , but determined only with εq n precision. In the next theorem we show that this precision can be improved all the way to an n-independent constant. We have Theorem 7.1. Fix κ, ν, ǫ. Then for sufficiently small ε there existsn 0 (κ, ν, λ, α, ǫ, ε) such that if θ is k-Diophantine for alln 0 ≤ k ≤ n with Diophantine constants κ, ν and ln qn+1−ln j qn > ǫ ln λ with ǫ > 0, then Proof. We first give the proof of (101). Let k 0 ∈ [jq n − εq n , jq n ] be such that Suppose (101) does not hold, i.e., k 0 ∈ [jq n − εq n , jq n − K 0 ]. Now we will reflect the elements in [jq n − εq n , jq n ] at j 2 q n . That is for any element k ∈ [jq n − εq n , jq n ], let k ′ = jq n − k. Then k ′ ∈ [0, εq n ]. Choose , Notice that I ′ 2 and I 2 are reflections of each other about j 2 q n . By the Diophantine condition on θ with respect to α, for any k 2 ∈ I 2 (k ′ 2 ∈ I ′ 2 ) and k 1 ∈ I 1 , we have where the last inequality holds by the fact |k 1 |, |k ′ 2 | ≤ C ⋆ b n ′ +1 so that we can apply Lemma B.2.
For any k 2 ∈ I 2 and k 1 ∈ I 1 (k 1 + k ′ 2 = 0 by the construction of I 1 , I 2 ), we also have where the last inequality holds by the fact |k 1 |, |k ′ 2 | ≤ C ⋆ b n ′ +1 and k 1 − k ′ 2 = q n ′ Z so that we can apply Lemma B.3.

By Theorem B.4, (103) and (104), we have
Similarly, This is impossible. (103) and (104), we know that the small divisor condition does not change under reflection at j 2 q n . Following the proof of (54) and replacing Lemma 4.1 with a combination of Lemma 4.2 and Theorem B.4, we have This is contradicted to the fact that k 0 is the maximal point because |k ′ 0 − ℓq n ′ | ≤ b n ′ . This completes the proof of (101). Now we turn to the proof of (102). Let k r 0 ∈ [jq n , jq n + εq n ] be such that ||U (k r 0 )|| = sup k∈[jqn,jqn+εqn] ||U (k)||.
Suppose the Theorem does not hold, i.e., k r 0 ∈ [jq n + K 0 , jq n + εq n ]. In this case we shift the elements in [jq n , jq n + εq n ] by −jq n . That is for any element k ∈ [jq n , jq n + εq n ], let k r,′ = k − jq n . Then k r,′ ∈ [0, εq n ]. Then (102) holds by the same proof, only replacing all k ′ with k r,′ .
Proof. By shifting the operator by k 0 units, we can assume k 0 = 0. Theorem 7.1 still holds if 0 is a local (n + 1)-maximum by Remark 6.5.
Moreover, every local n j−s -maximum on the interval b an j ,an j−1 ,...,an j−s+1 + [−e ǫ ln λqn j−s , e ǫ ln λqn j−s ] is of the form b an j ,an j−1 ,...,an j−s for some a nj−s .
First we prove the case s = 1. By the Diophantine condition on θ (109), we have for |ℓ| ≤ q nj−1+1 , the following holds Therefore θ + b an j α isn 0 + 1-Diophantine with respect to α with parameters 3ν, κ, and by Theorem 7.2 again, there exists a local n j−1 -maximum such that b an j ,an j−1 = a nj−1 q nj−1 + b an j +K nj−1 with |K nj−1 | ≤ K 1 = qn 0+2 . This completes the first step.
Assume Theorem holds for s = k − 1. It suffices to show it holds for s = k. By the Diophantine condition on θ (109) again, we have for |ℓ| ≤ q n j−k +1 , the following holds, ||2θ + (2b an j ,an j−1 ,··· ,an j−k+1 Thus θ + b an j ,an j−1 ,··· ,an j−k+1 α isn 0 + k-Diophantine with respect to α with parameters 3ν, κ, and by Theorem 7.2 again, there exists a local n j−k -maximum such that b an j ,an j−1 ,··· ,an j−k = a n j−k q n j−k + b an j ,an j−1 ,··· ,an j−k+1 +K n j−k with |K n j−k | ≤ K k = qn 0 +k+1 . This implies II holds for s = k. Thus we complete the proof of I and II.
III, as well as the moreover part, follow from Theorem 2.2 directly.
Proof of Theorem 2.4 Proof. Since k 0 is a local n j + 1-maximum for every j, Theorem 2.4 follows from Theorem 7.3 directly.
Theorem 7.3 describes a hierarchical structure around every local (n j + 1)-maximum. We will say that a local n-maximum is k-hierarchical if there exists ǫ > 0, n 1 , . . . , n k with n i+1 > e ǫni , n > e ǫn k , and, for each s = 1, . . . , k, a collection of local n s -maxima, {b an 1 ,...,an s } such that Proof of Corollary 2.7. Clearly, b an 1 ,...,an s of Theorem 7.3 form the collection required for the definition of k-hierarchy, so it remains to estimate the number of levels of the hierarchy, that is find k such that n j−k ≥n 0 + k. Clearly, k = j/2 − ⌊n 0 /2⌋ works. 8. Growth of transfer matrices. Proof of (13) Assume θ is Diophantine with respect to α in this and next section.
Theorems 8.1 and 8.2 imply the following theorem directly.

Proof of the corollaries
Proof of Corollary 2.12 Proof. Due to (28), i) follows from iii). By Theorem 2.1 to prove iii), it is enough to show that for any ε > 0, sufficiently large n and εq n < k < q n , we have Let m ≤ n + 1 be such that q m /2 ≤ k < q m+1 /2. If q m ≤ k, we are in Case 1 of the definition of g with m ≤ n − 1. Notice that ℓq m ≥ εq n ≥ εq m+1 , which leads to ln q m+1 ℓ qm being small. Then (135) follows from (6). If q m /2 < k < q m (135) is automatic by the definition of g.
It remains to establish ii). First by (98), we must have Let j k = ⌊q ε n k +1 ⌋, where sequence q n k is given by (99). Then Combining with Theorem 2.1, we must have ii) holds by (136) and (137).
To prove iv) it suffices to show that for any q nj − β 2 ln λ q nj + εq nj ≤ k ≤ q nj − εq nj ,we have ||U (k)|| ≥ e −(ln λ−cε)k where q nj is a subsequence satisfying (99).Indeed, under this assumption we are in Case 1 of the definition of f and the second addend dominates in (7) leading to the statement.

Remarks
• If we take for q n k a subsequence with any bounded away from zero exponential growth, we still get non-Lyapunov behavior on intervals of the form [q n k −cq n k +εq n k , q n k −εq n k ] for some c < β 2 ln λ .
• In fact, in all the arguments β can be replaced with ln q n+1 /q n .
Combining with (139), we must have This implies (140) and also implies (21). Now we verify (22). By the definition of f (k), g(k), for any large k, we have Then by (139) and Theorem 2.1 again, one has Let k j = q nj . One has Combining with (139) and Theorem 2.1 again, we get (145) lim j→∞ ln δ kj k j = −β.
We are now ready to study the behavior at non-resonant points. For an n-nonresonant y, let, as before, n 0 be the least positive integer such that 4q n−n0 ≤ dist(y, q n Z). Let s be the largest positive integer such that 4sq n−n0 ≤ dist(y, q n Z). Recall that, automatically, n 0 ≤ C(α). Set I 1 , I 2 ⊂ Z as follows ln | sin π(2θ + (j + i)α)| ≥ −C ln q n . and (157) min ln | sin π(j − i)α)| ≥ −C ln q n .
Proof of Theorem 3.5.
In case i), (156) and (157) are obtained correspondingly from Lemmas B.2 and B.3, thus Theorem 3.5 follows from Theorem B.4. In case ii) it is easy to see that (156) and (157) also hold, so Theorem B.4 applies as well.
By (149) and (150), the corresponding minimum term of min m∈I1∪I2,m =i ln | sin π((m−i)α)| is achieved at jq n . It is easy to check that