Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schr\"odinger operators

We introduce a notion of $\beta$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $\beta$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schr\"odinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.


Introduction
Singular continuous spectral measures of Schrodinger operators, usually defined by what they are not, are still not very well understood. The aim of direct spectral theory is to obtain properties of spectral measures/spectra and associated quantum dynamics based on the properties of the potential. In the context of 1D operators this is most often done via the study of solutions/transfer matrices/dynamics of transfer-matrix cocycles. Indeed, there are many beautiful results linking the latter to either dimensional properties of spectral measures (going back to [37]) or directly to quantum dynamics (e.g. [46,25]). There is also a long thread of results relating dimensional properties of spectral measures to quantum dynamics (e.g. [9,7] and references therein) as well as results connecting spectral/dynamical properties to some further aspects (e.g. [47,13]). Many of those have been used to obtain dimensional/quantum dynamical results (sometimes sharp) for several concrete families (e.g. [23]). However, there were no results directly linking easily formulated properties of the potential to dimensional/quantum dynamical results, other than for specific families or a few that ensure either the mere singularity or continuity of spectral measures (and their immediate consequences). In particular, we don't know of any quantitative results of this type. In this paper we prove the first such result. Consider Schrödinger operator on l 2 (Z) given by (Hu) n = u n+1 + u n−1 + V (n)u n (1.1) For β > 0, we say a real sequence {V (n)} n∈Z has β-repetitions if there is a sequence of positive integers q n → ∞ such that max 1≤j≤qn |V (j) − V (j ± q n )| ≤ e −βqn (1.2) We will say that {V (n)} n∈Z has ∞-repetitions if (1.2) holds for any β > 0. For β < ∞, we will say that {V (n)} n∈Z is β-almost periodic, if, for some ǫ > 0, V (· + kq n ) satisfies (1.2) for any |k| ≤ e ǫβqn /q n , i.e., max 1≤j≤qn,|k|≤e ǫβqn /qn |V (j + kq n ) − V (j + (k ± 1)q n )| ≤ e −βqn (1.3) for any n. We will say that {V (n)} n∈Z is ∞-almost periodic, if it is β-almost periodic for any β < ∞. We note that β and even ∞-almost periodicity does not imply almost periodicity in the usual sense. In particular, it is easily seen that there is an explicit set of generic skew shift potentials that satisfy this condition. We will prove Theorem 1 Let H be given by (1.1) and V is bounded and β-almost periodic. Then, for an explicit C = C(ǫ, V ) > 0, for any the spectral measure is γ-spectral continuous.
For the definition of spectral continuity (a property that also implies packing continuity and thus lower bounds on quantum dynamics) see Section 1.1. We formulate a more precise (specifying the dependence of C on ǫ, V ) version in Theorem 6.
Our result can be viewed as a quantitative version simultaneously of two well known statements • Periodicity implies absolute continuity. Indeed, we prove that a quantitative weakening (β-almost periodicity) implies quantitative continuity of the (fractal) spectral measure.
• Gordon condition (a single/double almost repetition) implies continuity of the spectral measure. Indeed, we prove that a quantitative strengthening (multiple almost repetitions) implies quantitative continuity of the spectral measure.
Potentials with ∞-repetitions are known in the literature as Gordon potentials 1 . This property has been used fruitfully in the spectral theory in various situations, see reviews [17,19] and references therein. In many cases those potentials were automatically β or even ∞almost periodic, so satisfied almost repetitions over sufficiently many periods. However, even in such cases, what all those papers used was the strength of the approximation over one-two (almost) periods based on Gordon Lemma type arguments. Our main technical accomplishment here is that we find a new algebraic argument and develop technology that allows to obtain quantitative corollaries from the fact that the approximation stays strong over many periods, thus exploring this feature analytically for the first time.
Lower bounds on spectral dimension lead to lower bounds on packing dimension, thus also for the packing/upper box counting dimensions of the spectrum as a set and for the upper rate of quantum dynamics. Therefore we obtain corresponding non-trivial results for all above quantities.
It is clear that our general result only goes in one direction, as even absolute continuity of the spectral measures does not imply β-almost periodicity for β > 0.
However, in the important context of analytic quasiperiodic operators this leads to a sharp if-and-only-if result.
Let H = H θ,α,V be a Schrödinger operator on l 2 (Z) given by (Hu) n = u n+1 + u n−1 + V (θ + nα)u n , n ∈ Z, θ ∈ T (1. 5) where V is the potential, α ∈ R\Q is the frequency and θ ∈ T is the phase. Let µ = µ θ,α be the spectral measure associated with vectors δ 0 , δ 1 ∈ l 2 (Z) in the usual sense. Given α ∈ (0, 1), let p n /q n be the continued fraction approximants to α. Define is the spectrum of H and L(E) is the Lyapunov exponent, be the set of supercritical energies (or, equivalently, the set of E such that the corresponding transfer-matrix cocycle is non-uniformly hyperbolic). S depends on α and V but not on θ.
Our main application is Theorem 2 For any analytic V and any θ, the spectral measure µ restricted to S is of full spectral dimension if and only if β(α) = ∞.
Full spectral dimensionality is defined through the boundary behavior of Borel transform of the spectral measure (see details in Section 1.1). It implies a range of properties, in particular, maximal packing dimension and quasiballistic quantum dynamics. Thus our criterion links in a sharp way a purely analytic property of the spectral measure to arithmetic property of the frequency. The result is local (so works for any subset of the supercritical set, see Theorem 4 for more detail) and quantitative (so we obtain separately quantitative spectral singularity and spectral continuity statements for every finite value of β, see Theorems 5 and 6).
The study of one-dimensional one-frequency quasiperiodic operators with general analytic potentials has seen remarkable advances in the last two decades, from the Eliasson's KAM point spectrum proof for the general class [27], to Bourgain-Goldstein's nonperturbative method [11], to Avila's global theory [2]. In particular, many results have been obtained in the regime of positive Lyapunov exponents (dubbed supercritical in [2]). They can be divided into two classes • Those that hold for all frequencies (e.g. [38,12,25,41,42,43]) • Those that have arithmetic (small denominator type) obstructions preventing their holding for all frequencies thus requiring a Diophantine type condition (e.g. [11,33,22]) 2 Results of the first kind often (but not always [12,59]) do not require analyticity and hold in higher generality. Results of the second kind describe phenomena where there is a transition in the arithmetics of the frequency, thus an extremely interesting question is to determine where does this transition happen and to understand the neighborhood of the transition. However, even though some improvements on the frequency range of some results above have been obtained (e.g. [60]), most existing proofs often require a removal of a non-arithmetically defined measure zero set of frequencies, thus cannot be expected to work up to the transition. There have been remarkable recent advances in obtaining complete arithmetic criteria in presence of transitions [6,39,40] or non-transitions [3] for explicit popular Hamiltonians: almost Mathieu operator and Maryland model, but there have been no such results that work for large families of potentials. Theorem 2 is the first theorem of this kind. A natural way to distinguish between different singular continuous spectral measures is by their Hausdorf dimension. However Hausdorff dimension is a poor tool for characterizing the singular continuous spectral measures arising in the regime of positive Lyapunov exponents, as it is always equal to zero (for a.e. phase for any ergodic case [57], and for every phase for one frequency analytic potentials [38] 3 ). Similarly, the lower transport exponent is always zero for piecewise Lipshitz potentials [25,43]. Thus those two quantities don't even distinguish between pure point and singular continuous situations. In contrast, our quantitative version of Theorem 2, contained in Theorems 5 and 6, shows that spectral 2 Not all results can currently be classified this way, most notable example being the Cantor structure of the spectrum [34], currently proved for a non-arithmetically-defined full measure set of frequencies, while the statement has no known arithmetic obstructions. Theoretically there may also be results such as [3] which formally should belong to the first group but the proof requires argument that highly depends on the arithmetics, so they must be in the second group, in spirit. In some sense [12] is a result of this type. 3 The result of [38] is formulated for trigonometric polynomial v. However it extends to the analytic case -and more -by the method of [43].
dimension is a good tool to finely distinguish between different kinds of singular continuous spectra appearing in the supercritical regime for analytic potentials.
The continuity part of Theorem 2 is robust and only requires some regularity of V . Besides the mentioned criterion, Theorem 1 allows us to obtain new results for other popular models, such as the critical almost Mathieu operator, Sturmian potentials, and others.
Indeed, our lower bounds are effective for β > C sup E∈σ(H) L(E) where L(E) is the Lyapunov exponent (see Theorem 6) thus the range of β is increased for smaller Lyapunov exponents, and in particular, we obtain full spectral dimensionality (and therefore quasiballistic motion) as long as β(α) > 0, when Lyapunov exponents are zero on the spectrum. This applies, in particular, to Sturmian potentials and the critical almost Mathieu operator.
As an example, setting S 0 = {E : L(E) = 0} we have Theorem 3 For Lipshitz V , the quantum dynamics is quasiballistic A similar statement also holds for full spectral dimensionality or packing/box counting dimension one. The Lipshitz condition can be relaxed to piecewise Lipshitz (or even Hölder), leading to part 1 also holding for Sturmian potentials. This in turn leads to first explicit examples of operators whose integrated density of state has different Hausdorff and packing dimensions, within both the critical almost Mathieu and Sturmian families.
The fact that quantum motion can be quasiballistic for highly Liouville frequencies was first realized by Last [49] who proved that almost Mathieu operator with an appropriate (constructed step by step) Liouville frequency is quasiballistic. Quasiballistic property is a G δ in any regular (a-la Simon's Wonderland theorem [56]) space [30,16], thus this was known for (unspecified) topologically generic frequencies. Here we show a precise arithmetic condition on α depending on whether or not Lyapunov exponent vanishes somewhere on the spectrum. Thus, in the regime of positive Lyapunov exponents, the quantum motion is very interesting, with dynamics almost bounded along some scales [43] (this property is sometimes called quasilocalization) and almost ballistic along others. For finite values of β(α) in this regime our result also yields power-law quantum dynamics along certain scales while bounded along others.

Main application
Fractal properties of Borel measures on R are linked to the boundary behavior of their Borel transforms [26]. Let be the Borel transform of measure µ. Fix 0 < γ < 1. If for µ a.e. E, we say measure µ is (upper) γ-spectral continuous. Note that spectral continuity (and singularity) captures the lim inf power law behavior of M (E + iε), while the corresponding lim sup behavior is linked to the Hausdorff dimension [26]. Define the (upper) spectral dimension of µ to be s(µ) = sup γ ∈ (0, 1) : µ is γ-spectral continuous .  we say measure µ is (upper) γ-spectral singular. We can also consider Obviously, s(µ) ≤ s(µ). The main theorem also holds for s(µ).

Spectral singularity, continuity and proof of Theorem 4
We first study γ-spectral singularity of µ. We are going to show that under the assumption of Theorem 4 we have: Theorem 5 Assume L(E) > a > 0 for E ∈ S. There exists c = c(a) > 0 such that for any α, θ, if (1.12) then µ S is γ-spectral singular.
Obviously, Theorem 5 implies that if β < +∞, then The analyticity of potential and positivity of Lyapunov exponent are only needed for spectral singularity. We now formulate a more precise version of the general spectral continuity result, Theorem 1.
For S ⊂ σ(H) assume there are constants Λ > 0 and n 0 ∈ N such that for any k ∈ Z, E ∈ S and n ≥ n 0 Clearly, such Λ always exists for bounded V, with n 0 = 1. As before we denote µ S the spectral measure of H restricted to a Borel set S ⊂ σ(H).
We will now get the lower bound using Theorem 6. Let V θ (n) := V (θ + nα). By boundedness of V and compactness of the spectrum, there is a constant Λ V < ∞ such that (1.14) holds uniformly for E ∈ σ(H θ ), θ ∈ T. In order to apply Theorem 6, it is enough to show that for any β < β(α), V (θ + jα) has β-repetitions for any θ ∈ T, j ∈ Z. Indeed, by (1.6), there is a subsequence q n k such that log q n k +1 q n k > β Since V is analytic, for any θ, j and 1 ≤ n ≤ q n k Thus if β(α) = ∞, s(µ S ) = s(µ S ) = 1.
Property (1.14) naturally holds in a sharp way in the context of ergodic potentials with uniquely ergodic underlying dynamics. Assume the potential V = V θ is generated by some homeomorphism T of a compact metric space Ω and a function f : Ω → R by V θ (n) = f (T n θ), θ ∈ Ω, n ∈ Z. (1.17) Assume (Ω, T ) is uniquely ergodic with an ergodic measure ν. It is known that the spectral type of H θ is ν-almost surely independent of θ (e.g [14]). In general, however, the spectral type (locally) does depend on θ ( [44]). If f is continuous then, by uniform upper-semicontinuity (e.g. [29]) This was recently extended in [43] to almost continuous f. Following [43], we will say a function f is almost continuous if it is bounded and its set of discontinuities has a closure of ν measure zero. By Corollary 3.2 in [43], if f is bounded and almost continuous then (1.18) also holds for every E. Moreover, if the Lyapunov exponent L(E) is continuous on some compact set S, then, by compactness and subadditivity, the lim sup in (1.18) will be also uniform in E ∈ S. Since by upper semicontinuity L(E) is continuous on the set where it is zero, as a consequence of Theorem 6, we obtain Corollary 1 Assume the function f in (1.17) is bounded and almost continuous and L(E) = 0 on some Borel subset S of σ(H θ ). If V θ (n) is β-almost periodic for some β > 0, ǫ > 0, then s(µ θ S ) = 1. Proof: For any 0 < γ < 1, set Λ ′ = β(1 − γ)/2C where C = C(ǫ) is given in Theorem 6. 4 Since L(E) = 0 on S, by the arguments above, there is n 0 = n 0 (Λ ′ ) independent of θ and E such that Obviously, β > CΛ ′ /(1 − γ), so Theorem 6 is applicable and (1.16) holds. Therefore, As an immediate consequence we obtain Theorem 7 If V θ (n) is given by (1.17) with uniquely ergodic (Ω, T ) and almost continuous f, then, for every θ we have 1. s(µ S 0 ) = 1, as long as V is β-almost periodic with β > 0.
is not a necessary condition in general for s(µ S 0 ) = 1, for s(µ ac ) = 1 even if V is not β-almost periodic for any β, and the support of the absolutely continuous spectrum is contained in (and may coinside with) S 0 . It is a very interesting question to specify a quantitative almost periodicity condition for s(µ sing S 0 ) = 1, in particular, find an arithmetic criterion for analytic one frequency potentials for s(µ Scr ) = 1 where S cr ⊂ S 0 is the set of critical energies in the sense of Avila's global theory.
2. According to Theorem 4, β = ∞ is also necessary if f is analytic and T is an irrational rotation of the circle (β will depend on T ). In case f has lower regularity, it is an interesting question to determine optimal condition on β.

Relation with other dimensions; Corollaries for the AMO, Sturmian potentials, and Transport exponents.
If we replace the lim inf by lim sup in the definition of upper spectral dimension, we will define correspondingly the lower spectral dimension which will coincide with the Haurdorff dimension dim H (µ) of a measure µ. Also one can consider the packing dimension of µ, denoted by dim P (µ). The packing dimension can be defined in a similar way as in (1.9) through the γ-dimensional lower derivative D γ µ(E). It can be easily shown that D γ µ(E) ≤ lim inf ε↓0 ε 1−γ |M (E+iε)|. Thus the relation between packing dimension and upper spectral dimension is dim P (µ) ≥ s(µ). 5 Therefore, the lower bound we get in Theorem 6 also holds for the packing dimension.
Lower bounds on spectral dimension also have immediate applications to the lower bounds on quantum dynamics. Denote by δ j be the vector δ j (n) = χ j (n). For p > 0, define The growth rate of |X| p δ 0 (T ) characterizes how fast does e −itH δ 0 spread out. In order to get the power law bounds for |X| p δ 0 (T ), it is natural to define the following upper β + δ 0 (p) and lower β − δ 0 (p) dynamical exponents as The dynamics is called ballistic if β − δ 0 (p) = 1 for all p > 0, and quasiballistic if β + δ 0 (p) = 1 for all p > 0. We will also say that the dynamics is quasilocalized if β − δ 0 (p) = 0 for all p > 0.
In [35], it is shown that the upper and lower transport exponents of a discrete Schrödinger operator (1.1) can be bounded from below by the packing and Hausdorff dimension of its spectral measure respectively. Therefore, by [35] we have β + δ 0 (p) ≥ s(µ), ∀p. As a direct consequence of Theorem 6 we have Corollary 2 If V (n) is bounded and ∞-almost periodic, the upper dynamical exponent β + δ 0 (p) of the operator (1.5) is one for any p > 0, and the associated dynamics is quasiballistic.
For λ > 1, Hausdorff dimension of the spectral measure of the almost Mathieu operator is equal to zero [38] and β − (p) = 0 for all p > 0 [25]. Thus almost Mathieu operators with λ > 1 and β(α) = ∞ provide a family of explicit examples of operators that are simultaneously quasilocalized and quasiballistic and whose spectral measures satisfy 0 = dim H (µ) < dim P (µ) = 1. 6 And has spectral dimension one and packing dimension one of the spectral measure The same holds of course for cos replaced with any almost continuous f as long as Lyapunov exponent is positive everywhere on the spectrum, in particular for f = λg where g is either bi-Lipshitz (as in [36]) or analytic, and λ > λ(g) is sufficiently large.
Let dN be the density states measure of the almsot Mathieu opeartor and Σ be the spectrum. It is well known that in the critical case, λ = 1, Σ has Lebesgue measure zero ( [5,48]). It is then interesting to consider the fractal dimension of the spectrum (as a set). Since dN = E(dµ θ ) and supp top (dN ) = Σ , by the discussion above we have Corollary 4 For the critical almost Mathieu operator, λ = 1, and β(α) > 0 we have dim P (dN ) = dim P (Σ) = 1.
Last and Shamis proved in [51] (see also [53]) that for a dense G δ set of α (which therefore has a generic intersection with the set {α : β(α) > 0}), the Hausdorff dimension of the spectrum is equal to zero. Thus the spectrum of the critical almost Mathieu operator with a topologically generic frequency is an example of a set such that Moreover, Last [48] showed that if q n+1 > Cq 3 n for all n, (which is the set containing {α : β(α) > 0}) then dim H (Σ) ≤ 1/2. Thus critical almost Mathieu operators with β(α) > 0 and any θ provide an explicit family of operators that all have spectra satisfying We note that the question of fractal dimension of the critical almost Mathieu operator attracted a lot of attention in Physics literature, with many numerical and heuristic results. In particular, Wilkinson-Austin [58] conjectured that dim B (Σ) < 1/2 for all α and there were many results rigorously or numerically confirming this for certain α. Our corollary 4 provides an explicit example disproving this conjecture.
Another well known family are Sturmian Hamiltonians given by , it is called the Fibonacci Hamiltonian. The spectral properties of the Fibonacci Hamiltonian have been thoroughly studied in a series of papers in the past three decades, see [18,20] for more references. Recently, Damanik, Gorodetski, Yessen proved in [23] that for every λ > 0, the density of states measure dN λ is exact-dimensional (the Hausdorff and upper box counting dimension are the same) and Our results show that the exact dimensionality properties of Sturmian Hamiltonians strongly rely on the arithmetic properties of α. It was shown in [8] that if α is irrational, the Lyapunov exponent of Sturmian operator restricted to the spectrum is zero. Also the spectrum of Sturmian Hamiltonian Σ λ,α is always a Cantor set with Lebesgue measure zero. Moreover, for Sturmian potentials results similar to thoses for the critical Mathieu operator in Corollary 4 also hold. Let µ θ be the spectral measure of Sturmian operator (1.22) and let dN λ,α be the density states of measure and Σ λ,α be the spectrum. We say that phase θ is α-Diophantine if there exist γ < ∞, τ > 1 such that θ+mα R/Z ≥ γ (|m|+1) τ for all m ∈ Z. Clearly, this is a full measure condition. We have Theorem 8 For Sturmian operator H θ,λ,α with β(α) > 0 and λ > 0, if θ is α-Diophantine, the spectral dimension of µ θ is one.
Previously, Liu, Qu and Wen [55,54] studied the Hausdorff and upper box counting dimension of Σ λ,α of Sturmian operators. For large couplings, they gave a criterion on α ∈ (0, 1) for the Hausdorff dimension of the spectrum to be equal to one. Combining Theorem 8 with their results, we have Corollary 5 Let Σ λ,α be the spectrum of the Sturmian Hamiltonian with λ > 20. There are explicit α such that for a.e. θ, The proof for the Sturmian case is given in Section 4.
The rest of this paper is organized in the following way. After giving the preliminaries in Section 1.4 we proceed to the proof of the general continuity statement in Section 2. First we quickly reduce Theorem 6 to Lemma 2.1 where we also specify the constant C 0 appearing in Theorem 6. We note that we do not aim to optimize the constants here and many of our arguments have room for corresponding improvement. Lemma 2.1 is further reduced to the estimate on the traces of the transfer matrices over eventual almost periods, Theorem 9, through its corollaries, Lemmas 2.2 and 2.3. Theorem 9 is the key element and the most technical part of the proof. It is of interest in its own right as can be viewed as the quantitative version of the fact that period length transfer matrices of periodic operators are elliptic: it provides quantitative bounds on the traces of transfer matrices over almost periods based on quantitative almost periodicity, for spectrally a.e. energy. In section 2.2 we separate this statement into hyperbolic and almost parabolic parts, correspondingly Lemmas 2.4 and 2.5. In section 2.3 we use the extended Schnol's Theorem to study the hyperbolic case and in section 2.4 we combine estimates on level sets of the polynomials, power-law subordinacy bounds, and an elementary but very useful algebraic representation of matrix powers (Lemma 2.9) to study the almost parabolic case. Lemmas 2.2 and 2.3 are proved in Section 2.5, completing the continuity part. In Section 3 we focus on the analytic quasiperiodic potentials and prove Theorem 5. The proof is based on a lemma about density of localized blocks (Lemma 3.3). Finally, we discuss Sturmian potentials in Section 4, proving Theorem 8 and then providing explicit examples for Corollary 5.

m-function and subordinacy theory
In this part, we will briefly introduce the power-law extension of the Gilbert-Pearson subordinacy theory [31,32], developed in [37]. We will also list the necessary related facts on the Weyl-Titchmarsh m-function. More details can be found, e.g., in [14].
Let H be as in (1.5) and z = E + iε ∈ C. Consider equation (1. 25) with the family of normalized phase boundary conditions: Now given any ε > 0, we define lengths l = l(ϕ, ε, E), by requiring the equality We also define l − (ϕ) by u ϕ,− , v ϕ,− through the same equation. Direct computation shows that   [37]) For E ∈ R and ε > 0, the following inequality holds for any ϕ ∈ (− π 2 , π 2 ]: We need to study the whole-line m-function which is given by the Borel transform of the spectral measure µ of operator H (see e.g., [14]). The following relation between whole line m-function M and half line m-function m ϕ was first shown in [21] as a corollary of the maximal modulus principle. One can also find a different proof based on a direct computation in the hyperbolic plane in [4].
Proposition 1 (Corollary 21 in [21]) Fix E ∈ R and ε > 0, This proposition implies that in order to obtain an upper bound for the whole line mfunction, namely, the continuity of whole line spectrum, it is enough to obtain a uniform upper bound of the half line m-function for any boundary condition.
On the other hand, consider a unitary operator U : For any operator H on l 2 (Z), we define an operator H on l 2 (Z) by H = U HU −1 . Denote by m, m ϕ , u ϕ and l(ϕ), correspondingly, m, m ϕ , u ϕ and l(ϕ) of the operator H. We will need the following well known facts (see e.g. Section 3, [38]). For any ϕ ∈ (−π/2, π/2] we have Similar to Lemma 5 in [38], a direct consequence of relation (1.33) is the following result.

Transfer matrices and Lyapunov exponents
Although Theorem 6 does not involve any further conditions on the potential, it will be convenient in what follows to use the dynamical notations. Let Ω = R Z and T : Ω → Ω is given by (T θ)(n) = θ(n + 1). Let f (θ) := θ(0). Then any potential V can be written in the way (1.17), V θ (n) := θ(n) = f (T n θ). Thus for a fixed {V n } n∈Z = θ ∈ Ω, we will rewrite the potential V as V θ (n) = f (T n θ) as in (1.17). For our general theorem we do not introduce any topology, etc; this is being done purely for the notational convenience. Denote the n-step transfer-matrix by A n (θ, E): The connection to Schrödinger operators is clear since a solution of Hu = Eu can be reformulated as In other words, the spectral properties of Schrödinger operators H are closely related to the dynamics of the family of skew product (T, A(θ, E)) over Ω × R 2 . We will often suppress either θ or E or both from the notations if corresponding parameters are fixed through the argument. If V is actually dynamically defined by (1.17) with a certain underlying ergodic base dynamics (Ω, T, ν) then, by the general properties of subadditive ergodic cocycles, we can define the Lyapunov exponent 2 Spectral Continuity

Proof of Theorem 6
Throughout this section we assume (1.14) is satisfied uniformly for E ∈ S, θ ∈ Ω (see Section 1.4.2). Assume V is β-almost periodic for some ǫ > 0. The proof of Theorem 6 is based on the following estimates on the growth of the l-norm of the half line solutions. Let u ϕ , v ϕ be given as in (1.25)-(1.27). γ). For µ-a.e. E, there is a sequence of positive numbers η k → 0 so that for any ϕ where L k = ℓ k (ϕ, η k , E) is given as in (1.29).
The proof of Lemma 2.1 follows from the following estimates on the trace of the transfer matrix. Let q k be the sequence given in (1.3).

2)
then for µ a.e. E, there is K(E) such that This theorem is the key estimate of the spectral continuity. It can be viewed as a quantitative version of the classical fact that period-length transfer matrices of periodic operators are elliptic on the spectrum. Indeed, we prove that β-almost periodicity implies quantitative bounds on ellipticity. The proof will be given in the following two subsections. The direct consequence of Theorem 9 are the following estimates on the norm of the transfer matrices. They show that if the trace of the transfer matrix over an almost period is strictly less than 2, then the there is a sub-linearly bounded subsequence. We will use this result to prove Lemma 2.1 first. The proof of Lemma 2.2 will be left to Section 2.5. Let K(E) be given by Theorem 9. Then for µ-a.e. E, the following estimate holds: Additionally, Lemma 2.3 For 0 < γ < 1, assume that in addition to the conditions of Lemma 2.2 Proof of Lemma 2.1: It is enough to prove r.h.s. of (2.1) since then the l.h.s. of (2.1) follows from u ϕ L k v ϕ L k ≥ 1/4L k . For any 0 < γ < 1, set β 0 = 100(1 + 1/ǫ) Λ 1−γ , ξ = 17Λ 1−γ . Then (2.2),(2.4) and (2.6) are satisfied for all β > β 0 . Therefore, (2.7) holds with above choice of parameters. Let l k = [e ξq k ] · q k . Rewrite (2.7) as l k n=1 A n (E) 2 < l 2−γ k . Thus for any ϕ, v ϕ 2 l k ≤ 4l 2−γ k . By (1.30), we have Then, is well defined. Set L k (ϕ) := ℓ(ϕ, η k , E), the length scale satisfies By (2.10), Since η k ≤ ε k (ϕ) and u ϕ l v ϕ l is monotone increasing in l, we obtain for any ϕ By the definition of l k , for large k, where [L k (ϕ)] is the integer part of L k (ϕ). Define We have For large q k , it is easy to check Using (2.13), we have for any ϕ Together with the choice of β and ξ, we have β > β 0 > 15Λ + (2 + 1/ǫ)ξ k (ϕ) Now we can again apply Lemma 2.2 with parameters β, ξ k (ϕ) and the length scale N k (ϕ) = e ξ k (ϕ)·q k , to get By the l.h.s. of (2.17), we obtain

Proof of Theorem 9
The proof of Theorem 9 will be divided into two cases. We will first exclude the energies where the trace is much greater than 2 infinitely many times using extended Schnol's Theorem (Lemma 2.6). Then we will estimate the measure of energies where the trace is close to 2 through subordinacy theory. The conclusion consists of the following two lemmas. Again let q k be the sequence given by (1.3) with certain β, ǫ > 0. then for spectrally a.e. E, there is K 1 (E) such that,  [49]. An important step there was using Schnol's theorem to show that eventually spectrally almost every energy is in the union of the spectral bands of the periodic approximants enlarged by a factor of q 2 k . Here we show that spectrally almost every energy is in the union of the shrinked spectral bands of the periodic approximants, a much more delicate statement, technically, thus with more powerful consequences.

The hyperbolic case: Proof of Lemma 2.4
We are going to show that if q is an 'approximate' period as in (1.3) with certain β, ǫ > 0 and satisfies |Trace A q E | ≥ 2 + e −τ q (2.24) then the trace of the transfer matrix at the scale e τ q/2 will be very large and any generalized eigenfunction of Hu = Eu will be bounded from below at the scale e τ q/2 . If this happens for infinitely many q, then any generalized eigenfunction will have al least a larger than 1/2 power law growth (in index) along some fixed subsequence. By the extended Schnol's Theorem, such E must belong to a set of spectral measure zero.

Proof of Lemma 2.4:
Let q k be given as in (1.3). Combining Claim 1 with Lemma 2.6 with {x k } = ∪ i=4 i=1 {x i q k } we obtain that the set of E such that there are infinitely many q k j with |TraceA q k j E, α | ≥ 2 + e −τ q k j has spectral measure zero.
Claim 1 is based on the following results. First, we need to estimate the norm of the conjugation matrix for any hyperbolic SL(2, R) matrix w.r.t. the distance between its trace and 2: Lemma 2.7 Suppose G ∈ SL(2, R) with 2 < |TraceG| ≤ 6. The invertible matrix B such that where ρ ±1 are the two conjugate real eigenvalues of G with |detB| = 1 satisfies The proof is based on a direct computation of the conjugate matrices. For the sake of completeness, we present it in Appendix A.1.
Fix τ > 0 and apply Lemma 2.7 to A q satisfying We then have Claim 2 For large q, where ρ ±1 are the two conjugate real eigenvalues of A q with ρ > 1 and B satisfies |detB| Let us now finish the proof of Claim 1.
Lemma 2.6 is proved in the same way as the standard Schnol's Lemma, however the statement in this form, while very useful, does not seem to be in the literature (we learned it from S. Molchanov, see the Acknowledgement). For the sake of completeness, we include a short proof in the Appendix.

Energies with Trace close to 2: Proof of Lemma 2.5
All throughout this section, we will assume again that all q are large enough and satisfy (1.3) with certain β, ǫ > 0, i.e., max 1≤j≤q,|k|≤e ǫβq /q |V (j + kq) − V (j + (k ± 1)q)| ≤ e −βq , ǫ > 0. (2.42) We are going to show that spectrally almost surely, there are only finitely many q such that TraceA q is close to 2.
In fact, we are going to prove the following quantitative estimate on the measure of energies where the trace of the associated transfer matrix is close to 2.
Lemma 2.8 Let Λ be given by (1.14) on some set S ⊂ σ(H). Let where µ = µ S is the spectral measure restricted to S.
Once we have Lemma 2.8, Borel Cantelli lemma immediately implies Lemma 2.5. So the main task is to prove (2.45).
In order to estimate the spectral measure of S q , first we recall the following results on the structure of S q . Let P n (R) denote the polynomials over R of exact degree n. Let the class P n;n (R) be elements in P n (R) with n distinct real zeros.
Claim 5 Suppose E ∈ S q Σ and 0 < ε q < e (−τ /2+Λ/200)q . Suppose (2.42) holds. Assume that β > (2 + 1/ǫ)τ + (5 + 1/ǫ)Λ and τ ≥ 10Λ. Then for every initial condition ϕ, u ϕ 2 lq ≥ e Proof of Lemma 2.8: Take τ = 10Λ. Then β > (25 + 11/ǫ)Λ satisfies the requirement in Claim 5. Let E j ∈ I j Σ ⊂ S q Σ . For any ϕ, let u ϕ,E j , v ϕ,E j be the right half line solution associated with the energy E j . According to (2.49), Claim 5 can be applied to all u ϕ,E j . We have for any ϕ, u ϕ,E j 2 lq(j) ≥ e 1 10 Λq , j = 1, · · · , q where l q (j) = l(ϕ, E j , ε j q ). Then by the J-L inequality (1.31) and the definition of l q (j), we have Notice that the interval I j is independent of the boundary condition ϕ, and so is ε j q . Therefore, we can take the supremum w.r.t. ϕ on both sides of the above inequality. By Proposition 1, we have On the other hand, by the definition of M (z) in (1.7), we have Therefore, Since in (2.48) there are four cases for S q and each of them satisfies the previous estimates, the spectral measure of S q will be bounded by 4qe −Λq/15 ≤ e −Λq/20 .
The proof of Claim 5 relies on the following estimates on the transfer matrices. The first one is a formula for the power of a general SL(2, R) matrix. It is elementray but turned out particularly useful and will be an important part of our quantitative estimates in both hyperbolic and nearly parabolic cases. As we did not find it in the literature, we will provide a proof of it as well as of the next Lemma, in the Appendix. Lemma 2.9 Suppose A ∈ SL(2, R) has eigenvalues ρ ±1 . For any k ∈ N, if TraceA = 2, then Otherwise, A k = k(A − I) + I.
The key to the estimates in the nearly parabolic case is then the following simple Lemma 2.10 There are universal constants 1 < C 1 < ∞, c 1 > 1/3 such that for E ∈ S q and 1 ≤ k ≤ N q , we have Second, since A q (θ) is almost periodic (with an exponential error), the iteration of A q (θ) along the orbit will be close to its power. The argument is similar to what we used in the proof of (2.33) in the previous part.
According to Lemma 1.2, it is enough to find a ϕ such that both m ϕ and m π/2−ϕ are γ-spectral singular. The main technical tool to estimate m-function is the subordinacy theory Lemma 1.1. We also need one more general statement about the existence of generalized eigenfunctions with sub-linear growth in its l-norm (see [50]). That is, for µ θ -a.e. E, there exists ϕ ∈ (−π/2, π/2] such that u ϕ and u ϕ,− both obey lim sup This inequality provides us an upper bound for the l-norm of the solution. To apply subordinacy theory, we also need a lower bound for the l-norm. It will be derived from the following lower bounds on the norm of the transfer matrices. Denote We have There is c = c(a, ρ) > 0 such that for l > l(E, β, ρ), and any θ ∈ T, the following hold: Proof of Theorem 5: For any ϕ, we have Therefore, a direct consequence of (3.3) is the power law estimate for the left hand side of (3.5), i.e., u ϕ 2 l + v ϕ 2 l ≥ l 1+ 2c β for l large. On the other hand, according to (3.1), for µ θ -a.e. E, there exist ϕ(E) and C = C(E) < ∞, such that for large l, and then v ϕ l ≥ l 1/2+c/β (3.8) provided β < ∞ and l > l(β, E, ρ).
The proof of Lemma 3.1 depends on the following lemmas about the localization density of the half line solution. The key observation is that in the regime of positive Lyapunov exponents we can guarantee transfer matrix growth at scale q n somewhere within any interval of length q n , giving a contribution to (3.3).
There are c 2 = c 2 (a, ρ) > 0 and a positive integer d = d(ρ) such that for E ∈ S and n > n(E, ρ), there exists an interval ∆ n such that and for any θ ∈ ∆ n , we have 7 A n (θ, E, α) 2 HS > e nL(E)/8 (3.10) In the following, we will use · for the HS norm · HS . Now let c 2 and d be given as in Lemma 3.2. Denote where, as before q n are the denominators of the continued fraction approximants to α. Based on Lemma 3.2, one can show that Lemma 3.3 Fix E ∈ S and α ∈ R\Q. Let k n be given as in (3.11). Suppose q n is large enough so that (3.9) holds for ∆ kn . Then for any θ, and any N ∈ N, there is j N (θ) ∈ [2N q n , 2(N + 1)q n ) such that A j N (θ, E, α) > e knL(E)/16 (3.12) We first use Lemma 3.2 and Lemma 3.3 to finish the proof of Lemma 3.1. The proofs of these two lemmas are left to next section.
Proof of Lemma 3.1: For l sufficiently large, there is q n such that, l ∈ [2q n , 2q n+1 ). Write l as l = 2N q n + r, where 0 ≤ r < 2q n , 1 ≤ N < q n+1 qn . Suppose q n is large enough so that (3.9) holds for ∆ kn . Then Lemma 3.3 is applicable. Fix θ. Consider A n (θ, E, α) first. Let j n (θ) ∈ [2nq n , 2(n + 1)q n ), n = 0, 1, · · · , N, be given as in (3.12). Direct computation shows that ≥ N e knL(E)/16 7 We denote by · HS the Hilbert-Smith norm of a SL(2, R) matrix We have l = 2N q n + r < 4N q n , i.e., N > l/(4q n ). (3.11) implies c 2 q n /(5d) < k n < c 2 q n /(4d) for q n large, so we have where c = c(c 2 , d, a). Then for sufficiently large l, we have We also assume l large enough (meaning q n large enough), so that log q n+1 qn < 2β, i.e., For the same θ, repeat the above procedure for A n (θ−α, −α, E). Notice that A n (θ, E, α) = A n (θ−α, E, −α). Therefore, we have a sequence of positive integers j N (θ−α) ∈ [2N q n , 2(N + 1)q n ) for any N ∈ N such that The rest of the computations are exactly the same as for A n (θ, E, α). Notice that the constants c 2 and d in Lemma 3.2 are independent of the choice of α or −α and θ. So k n and c will be the same for A n and A n where C 1 = C 1 (S, V ρ ) can be taken uniform for all E ∈ S. Expand f n (θ) into its Fourier series on T as f n (θ) = k∈Z b n (k)e 2πikθ (3.16) where b n (k) is the k-th Fourier coefficients of f n (θ) so satisfies |b n (k)| < f n ρ · e −2πρ|k| , ∀k ∈ Z (3.17)

Proof of the density lemmas
We split f n (θ) into two parts, for some positive integer d which will be specified a little bit later For any θ ∈ T With this choice of d, we have 2πρd > C 1 + 1, so for any θ ∈ T |R n (θ)| ≤ 2 1 − e −2πρ e −n < 1, n > n 0 (ρ) (3.19) Now we assume that the Lyapunov exponent L(E) of A(θ, E) is positive. Denote and if g n (θ) > e nL(E)/4 , then f n (θ) > g n (θ) − |R n (θ)| > e nL(E)/8 , n > n(E) Therefore, we have for large n, On the other hand, Since g n (θ) is a trigonometric polynomial of degree 2dn, the set Θ 2 n consists of no more than 4dn intervals. Therefore, there exists a segment, ∆ n ⊂ Θ 2 n , with Leb(∆ n ) > c 2 4dn . Obviously, ∆ n is also contained in Θ 1 n , i.e., for any θ ∈ ∆ n , A n (θ) 2 HS > e nL(E)/8 and where d only depends on ρ and is independent of n.
The usual way to study Sturmian Hamiltonian is to decompose Sturmian potentials into canonical words, which obey recursive relations. Here we present an alternative approach to study spectral dimension properties of Sturmian Hamiltonian based on the technics we developed in Theorem 6.

A Appendix
A.1 Proof of Lemma 2.7: Suppose that u, v are the two normalized eigenvectors of G such that Gu = ρu, Gv = ρ −1 v, u = v = 1 Denote the angle between u and v by θ. Without loss of generality we assume further that |θ| < π/2. Set B = (u, v), B = A.2 Proof of Lemma 2.9 and Lemma 2.10 Proof of Lemma 2.9: Suppose A = a b c d ∈ SL(2, R) has eigenvalues ρ and ρ −1 .
Case I: TraceA = 2. Obviously, ρ = 1 and where B is the conjugation matrix. Suppose ρ = d. We can pick the conjugation matrix as If ρ = d, it is easy to see that bc = 0. Without loss of generality, we assume c = 0, b = 0. We can pick the conjugation matrix as Direct computation using (A.1),(A.2),(A.3) shows that for any k ∈ N, Case II: TraceA = 2. Also follows by a (simpler) direct computation, considering separately a = 1 and a = 1.
A.3 Some estimates on matrix products where M ≥ 1 only depends on N . Let G j = G + ∆ j , j = 1, · · · , N, be a sequence of two by two matrices with δ = max