Spectral Dimension for β -almost periodic singular Jacobi operators and the extended Harper's model
aa r X i v : . [ m a t h . SP ] A p r SPECTRAL DIMENSION FOR β -ALMOST PERIODIC SINGULAR JACOBIOPERATORS AND THE EXTENDED HARPER’S MODEL RUI HAN, FAN YANG AND SHIWEN ZHANG
Abstract.
We study fractal dimension properties of singular Jacobi operators. We prove quan-titative lower spectral/quantum dynamical bounds for general operators with strong repetitionproperties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positiveLyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensional-ity. The applications include the extended Harper’s model where we obtain arithmetic results onspectral dimensions and quantum dynamical exponents. Introduction
In this paper, we study self-adjoint Jacobi operators on ℓ ( Z ) given by:(1.1) ( Hu ) n = w n u n +1 + w n − u n − + v n u n , n ∈ Z where w n ∈ C \ { } and v n ∈ R are bounded sequences in n . If w n ≡ H is a discrete Schr¨odingeroperator. We will focus on singular Jacobi operators, where the off-diagonal sequence w n has anaccumulation point 0 at ±∞ . A prime example of such operator, both in math and in physicsliterature, is the extended Harper’s model (EHM), see (2.13).We are interested in the fractal decomposition of the spectral measure and quantitative spec-tral/quantum dynamical bounds. In a recent work of Jitomirskaya and Zhang [32], many quan-titative criteria of fractal dimensions of spectral measure for lattice Schr¨odinger operators wereobtained. Their criteria have various applications to quasiperiodic Schr¨ondiger operators, e.g., thealmost Mathieu operator or the Sturmian Hamiltonians. However, whether their results could beapplied to the Jacobi case, in particular the singular Jacobi case, has remained a question. Indeed,many generalizations of the spectral theory from the Schr¨odinger case to the singular Jacobi casehave shown to be highly nontrivial, e.g. [23, 28, 37, 22, 21, 3, 41].In this paper we give general sufficient conditions for spectral continuity in the singular Jacobicase, see Theorem 2.1. We show spectral continuity follows if the parameters w n , v n of H satisfy: (i)strong repetition properties; (ii) control of the averaged closeness between w n and 0. In particular,condition (ii) is imposed to control the strength of singularity. We also show such operators existwidely in the general context of quasiperiodic setting. In the positive Lyapunov exponent regimeof analytic quasiperiodic Jacobi operators, the general statement leads to the first arithmetic if-and-only-if criterion for full spectral dimensionality. Notably, our results have applications to theextended Harper’s model for both spectral and quantum dynamical properties.Our proof is based on a general dynamical system approach, which has recently shown to beextremely powerful in the study of spectral properties, e.g. [35, 5, 34, 17, 12, 24, 36]. The eigenvalueequation of (1.1), is associated to a linear cocycle system, see (3.3). In the Schr¨odinger case thecocycles are SL(2 , R )-valued, whereas in the singular Jacobi case the cocycles are GL(2 , C )-valuedwith determinants approaching zero along a sub-sequence. This presents the main obstruction in[23, 28, 37, 22, 21, 3, 41] and in our paper.It was shown in [32] that the fractal dimensions of spectral measures depend on the competitionbetween the quality of repetitions and the growth of the Schr¨odinger cocycles. Such competition was resolved in the SL(2 , R ) setting involving delicate algebraic arguments, which are difficult to carryout directly in the GL(2 , C ) setting due to the presence of singularity. To reduce to SL(2 , R ) case, weemploy a family of conjugacies which were first introduced in a recent work of Avila-Jitomirskaya-Marx [3]. Such regularization moves the singularity into the conjugate matrices. The main technicalaccomplishment of our work is to develop general quantitative estimates (see Lemmas 5.3, 5.4 and5.5) of the conjugacy under assumptions (i) and (ii). The successful combination of these estimateswith the mechanism in [32] proves the quantitative spectral continuity results for the singular Jacobicase.We show the assumptions (i) and (ii) hold for singular Jacobi operators over a quasiperiodicbase. In particular, the proof of (ii) is close in spirit to the characterization of singularity in [22](see also [27, 31]). Here we need to study the finer decomposition of the singular spectral measure,thus a strengthened characterization is developed. Moreover, our estimates hold for general C k sampling functions with finitely many non-degenerate zeros, which reduces the analytic regularityrequirements in [27, 31, 22]. This part is also of independent interest in the study of uniformupper-semi continuity of the Lyapunov growth.The rest of this paper is organized in the following way. In section 2, we give all the definitionsand state our main results. After giving the preliminaries in section 3, we proceed to discuss the(Λ , β ) bound in the quasiperiodic case in section 4. In section 5, we prove the general spectralcontinuity results. In section 6, we focus on the analytic quasiperiodic Jacobi operator and provearithmetic if-and-only-if criterion for full spectral dimensionality. In the last section, we discuss theexplicit parameter partitions for the extended Harper’s model.2. Main results
To formulate the main results, we introduce the following definitions.
Definition 2.1.
A sequence { a n } n ∈ Z is said be to β - q almost periodic if there exist δ > β > q ∈ N , such that the following holds: max | m |≤ e δβq | a m − a m ± q | ≤ e − βq . (2.1)We say { a n } n ∈ Z is β -almost periodic (about q n ) if there exists a sequence of positive integers q n → ∞ ,such that { a n } is β - q n almost periodic. Remark . The β -almost periodicity was first introduced in [32] to study quantitative spectralbounds in the Schr¨odinger case. Note that the β -almost periodicity does not imply the almostperiodicity in the usual sense. A typical example is the sequence generated by skew-shift map( x, y ) ( x + y, y + 2 α ) with a smooth sampling function f ( x, y ) on T . The sequence v n = f ( x + ny + n ( n − α, y + 2 nα ) is β -almost periodic for typical α , but not almost periodic for any α . Definition 2.2.
We say w n is (Λ , β )- q bounded if there exist Λ > , β > , δ >
0, and q ∈ N , suchthat min | m |≤ e δβ q m + q − Y j = m | w j | > e − Λ q . (2.2)We say w n is (Λ , β ) bounded (about q n ) if there exists a sequence of positive integers q n → ∞ , suchthat w n is (Λ , β )- q n bounded. Remark . If we only consider the maximum of (2.1) and (2.2) over | m | ≤ q n , then the standardGordon-type argument will be enough to show the absence of point spectrum for the associated Ja-cobi operator, provided β & Λ. Assume further the Lyapunov exponent is positve, then the operator
PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 3 has purely singular continuous spectrum by Kotani theory [33], see e.g. [8, 6]. See more discussionon the Gordon-type argument and purely singular continuous spectrum in [9] and references therein.Let µ be the spectral measure of the Jacobi operator given as in (1.1). The fractal properties of µ are closely related to the boundary behavior of its Borel transforms, see e.g. [13]. Let(2.3) M ( E + iε ) = Z d µ ( E ′ ) E ′ − ( E + iε )be the (whole line) Weyl-Titchmarsh m -function of H . We are interested in the following fractaldimension of µ : Definition 2.3.
We say µ is (upper) γ -spectral continuous if for some γ ∈ (0 ,
1) and µ a.e. E , wehave(2.4) lim inf ε ↓ ε − γ | M ( E + iε ) | < ∞ . Define the (upper) spectral dimension of µ to bedim spe ( µ ) = sup (cid:8) γ ∈ (0 ,
1) : µ is γ -spectral continuous (cid:9) . (2.5)Our first result is about spectral continuity and the lower bound on the spectral dimension. Theorem 2.1.
Let H be given as in (1.1) and let µ be the spectral measure of H . Assume thatthere are positive constants Λ , β, δ and a sequence of positive integers q n → ∞ such that w n , v n are β -almost periodic and w n is (Λ , β ) bounded about q n . There exists an explicit constant C = C ( δ, Λ , k w k ∞ , k v k ∞ ) > , such that if β > C and γ < − Cβ , then µ is γ -spectral continuous.Consequently, we have the following lower bound on the spectral dimension of µ : dim spe ( µ ) ≥ − Cβ . (2.6)We will formulate a more precise lower bound (specifying the dependence of C on δ, Λ , k w k ∞ , k v k ∞ )in Theorem 5.2.It is well known that periodicity implies absolute continuity. We actually prove a quantitativeweakening version of this result: β -almost periodicity implies γ -spectral continuity. On the otherhand, it is well known that Gordon condition implies absense of point spectrum, which predictspurely singular continuous spectrum in many situations. Our result distinguishes the singular con-tinuous spectrum further according to their spectral dimensions. This can be viewed as a quantitativestrengthening of Gordon-type results. Quantitative results directly linking easily formulated prop-erties of the potential to dimensional/quantum dynamical results were first proved in [32] for theSchr¨odinger case. Theorem 2.1 was a further generalization of this type of estimates to more generalsingular Jacobi operators.An important context where we have generic β -almost periodicity and (Λ , β ) bound is thequasiperiodic Jacobi operators with smooth sampling functions defined as follows. Consider realand complex valued sampling functions v : T R and c : T C . We also assume ln | c | ∈ L ( T ),which is the minimum requirement for the Lyapunov exponent to exist. Let H α,θ = H α,θ,c,v be theJacobi operator on ℓ ( Z ) given by:( H α,θ u ) n = c ( θ + nα ) u n +1 + ¯ c (cid:0) θ + ( n − α (cid:1) u n − + v ( θ + nα ) u n , n ∈ Z , (2.7)where θ ∈ T := [0 ,
1] is the phase, α ∈ [0 , \ Q and ¯ c ( θ ) is the complex conjugate of c ( θ ) in the usualsense. RUI HAN, FAN YANG AND SHIWEN ZHANG
Given α , let p n /q n be the continued fraction approximants to α . Define(2.8) β ( α ) := lim sup n ln q n +1 q n ∈ [0 , ∞ ] . It is easy to check that for any Lipschitz continuous sampling functions v and c , the sequences v ( θ + nα ) , c ( θ + nα ) are β -almost periodic as defined in (2.1) for any θ ∈ T and any β < β ( α ) / c , then c ( θ + nα ) will be (Λ , β ) bounded for a.e. θ . As a consequence ofTheorem 2.1, we have spectral continuity for a.e. θ for (2.7). More precisely, let µ α,θ be the spectralmeasure of H α,θ (2.7), we have: Corollary 2.2.
Assume v ( θ ) is Lipschitz continuous on T and c ( θ ) is C k continuous on T withfinitely many non-degenerate zeros . For all k = 1 , , · · · , there exists an explicit constant C = C ( c, v, k ) > and a full measure set Θ = Θ( α, c ) ( T , only depending on α and the zeros of c ( θ ) with the following properties: suppose β ( α ) > C , then for any θ ∈ Θ , (a): H α,θ has no eigenvalues in the spectrum; (b): the spectral dimension of µ α,θ is bounded from below as: dim spe ( µ α,θ ) ≥ − Cβ . (2.9)
In particular, if β ( α ) = ∞ , then for a.e. θ , dim spe ( µ α,θ ) = 1 . We will prove (Λ , β ) bound of c ( θ + nα ) for a.e. θ in section 4 and then part (b) follows directlyfrom Theorem 2.1. The main ingredient is one fundamental estimate (see Lemma 4.1) about thetrigonometric product over irrational rotation in [1]. Similar arguments have been used in [27, 22, 31]to study the arithmetic criterion of purely singular continuous spectrum. In those papers, the authorsconsidered periodic approximation based on Gordon-type arugment. The growth of the transfermatrix only need to be controlled within at most two periods. In our case, the quantitative spectralcontinuity relies on (Λ , β ) bound over exponentially many periods. The use of Lemma 4.1 is moredelicate and involved. See more details in Lemma 4.2.As mentioned before, the absence of point spectrum in part (a) is a direct consequence of the(Λ , β ) boundedness of c ( θ + nα ) and the standard Gordon-type argument. In view of Definition2.3, it is easy to check that point measure has zero (spectral/Haursdoff/packing) dimension. (2.9)implies that the spectral measure µ α,θ has positive spectral dimension for β > C . Part (a) can alsobe derived as a corollary of (2.9). An interesting question that remained here is whether the assump-tion on c ( θ ) can be weakened: For example, could any Lipschitz continuous function with finitelymany zeros generate a (Λ , β ) bounded sequence? Will the associated Jacobi operator have absenceof point spectrum and full spectral dimension? We will not go further in this direction in the cur-rent paper. We are planning to answer some of these questions in another paper (under preparation).It is clear that our general results (2.6) and (2.9) only go in one direction, as even absolutecontinuity of the spectral measures does not imply β -almost periodicity for β > . However, in theimportant context of analytic quasiperiodic operators (e.g. EHM) this leads to a sharp if-and-only-ifresult in the positive Lyapunov exponent regime.Let H α,θ be the Jacobi operator on ℓ ( Z ) defined as in (2.7). The Lyapunov exponent of H α,θ at energy E is defined through the associated skew-product over irrational rotations (quasiperiodiccocycles). For any irrational α , the Lyapunov exponent is only a function of E, α and is independentof θ , therefore, denoted as L ( E, α ). See more basic properties and discussions about Lyapunovexponent in section 3. We say θ ∈ T is a non-degenerate zero of f ∈ C k ( T , C ) if f ( θ ) = 0 and f ( k ) ( θ ) = 0. PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 5
Assume further v, c of H α,θ are analytic on T with real and complex values, respectively. Let µ α,θ, Σ + be the restriction of the spectral measure µ α,θ of H α,θ on Σ + := { E ∈ σ ( H α,θ ) : L ( E, α ) > } .We have the following sharp estimate on the spectral dimension of µ θ,α, Σ + : Theorem 2.3.
For any α ∈ [0 , , let β ( α ) be defined as in (2.8). For any analytic samplingfunctions v and c , there is a full Lebesgue measure set Θ = Θ( α, c ) ⊂ T explicitly depends on α and c such that for any θ ∈ Θ , dim spe (cid:0) µ α,θ, Σ + (cid:1) = 1 if and only if β ( α ) = ∞ . The proof of Theorem 2.3 contains two parts. Clearly, the ‘if’ part of Theorem 2.3 is a directconsequence of spectral continuity and follows from Corollary 2.2. The ‘only if’ part is usuallyreferred to as the so-called spectral singularity, defined through the singular boundary behavior ofthe m function. More precisely, we say the spectral measure µ is (upper) γ -spectral singular if forsome γ ∈ (0 ,
1) and µ a.e. E ,(2.10) lim inf ε ↓ ε − γ | M ( E + iε ) | = + ∞ . Define(2.11) dim f spe = inf (cid:8) γ ∈ (0 ,
1) : µ is γ -spectral singular (cid:9) . Obviously, dim spe ≤ dim f spe . Theorem 2.3 also holds for dim f spe . We actually can prove the followinglocal quantitative upper bound of the spectral dimension which completes the sufficient part ofTheorem 2.3. Theorem 2.4.
Consider the quasiperiodic Jacobi operator defined in (2.7) with analytic samplingfunctions v, c . Let L ( E ) be the associated Lyapunov exponent defined in (3.12). Assume L ( E ) ≥ a > on a compact set S . Consider the spectral measure µ α,θ restricted on S , denoted by µ α,θ,S . Suppose β ( α ) < ∞ , then there is a C = C ( a, v, c, S ) > and a full Lebesgue measure set Θ = Θ( α, c ) , suchthat for any θ ∈ Θ , µ α,θ is γ -spectral singular for any γ ≥ Cβ . Consequently, (2.12) dim spe ( µ α,θ, S ) ≤ dim f spe ( µ α,θ, S ) ≤
11 + C β < . The spectral singularity can be viewed as “weak-type of localization”. It involes the decay ofthe Green’s function in a finite box with a low density (see Lemma 6.4). Such decay/localizationdensity was previously known either with a strong non-resonance condition on ω (e.g. β ( ω ) = 0, see[12]), or for a concrete example with β ( ω ) . L (see [4, 26]). Such a phenomenon was first found in[32] for general analytic quasiperiodic Schr¨odinger operators with extremetely large β . Two crucialingredients for the quantitative spectral singularity are:(1) quantitative subordinate theory (Jitomirskaya-Last inequality, Lemma 3.2);(2) existence of generalized eigenfunctions with sub-linear growth by Last-Simon estimate (Lemma3.3).Theorem 2.4 generalizes the result for Schr¨odinger operators in [32] to singular Jacobi operators.The techniques to deal with the singular Jacobi case are more involved and very delicate in view ofthe quantitative estimates (2.12). In section 6, we reduce the proof of Theorem 2.4 to a quantitativeresult (see Lemma 6.1) obtained in [32]. One key observation in [32] is that the norm of the analytictransfer matrix can be approximated by trigonometric polynomials with uniform linear degree. Thegeneralization of this result to the meromorphic transfer matrix in our case (see Lemma 6.2) becomesan important part of the proof of Theorem 2.4. RUI HAN, FAN YANG AND SHIWEN ZHANG
Applications to the extended Harper’s model.
Quasiperiodic Jacobi operators arise nat-urally from the study of tight-binding electrons on a two-dimensional lattice exposed to a perpen-dicular magnetic field. A more general model is the extended Harper’s model (EHM), defined asfollow:(2.13) ( H λ,α,θ u ) n = c λ ( θ + nα ) u n +1 + ¯ c λ (cid:0) θ + ( n − α (cid:1) u n − + 2 cos 2 π ( θ + nα ) u n , n ∈ Z . Here, c λ ( θ ) = λ e − πi ( θ + α ) + λ + λ e πi ( θ + α ) , (2.14)¯ c λ ( θ ) is the complex conjugate of c λ ( θ ) in the usual sense and λ = ( λ , λ , λ ) ∈ R are real couplingconstants. EHM was introduced by D.J.Thouless in 1983 [40], which includes the AMO as a specialcase.The extended Harper’s model is a prime example of quasiperiodic Jacobi matrix. It has attractedgreat attention from both mathematics and physics (see e.g. [7, 10, 20]) literature in the past severaldecades. Recent developments on the spectral theory of the AMO and EHM include: pure pointspectrum for Diophantine frequencies in the positive Lyapunov exponent regime I o [23]; explicitformula for the Lyapunov exponent L ( E, λ ) (see (7.3)) on the spectrum throughout all the threeregions [28]; dry ten Martini problem for Diophantine frequencies in the self-dual regions [21]; com-plete spectral decomposition for all α and a.e. θ in the zero Lyapunov exponent regiems [3]; andarithmetic spectral transition in α in the positive Lyapunov exponent regime [22].As a central example of the analytic quasiperiodic singular Jacobi operators, Theorem 2.3 can beapplied to the extended Harper’s model H λ,α,θ defined in (2.13). As a consequence of the Lyapunovexponent formula of EHM in terms of the coupling constants λ = ( λ , λ , λ ), we have more explicitconclusions on the full spectral dimensionality of EHM. Moreover, our lower bounds in Theorem 2.3are effective for β > max { C sup E ∈ σ ( H ) L ( E ) , } by some simple scaling argument (see Lemma 4.2and section 7). Thus the range of β is increased for smaller Lyapunov exponents. In particular, weobtain full spectral dimensionality as long as β ( α ) > , when Lyapunov exponents are zero on thespectrum. This applies, in particular, to the critical EHM.Consider the following three parameter regions of λ = ( λ , λ , λ ) ∈ R : R = (cid:8) λ ∈ R : 0 < λ + λ < , < λ < (cid:9) . R = (cid:8) λ ∈ R : λ > max { λ + λ , } , λ + λ ≥ λ + λ > max { λ , } , λ = λ , λ > (cid:9) . R = { λ ∈ R : 0 ≤ λ + λ ≤ , λ = 1 or λ + λ ≥ max { λ , } , λ = λ , λ > } . Corollary 2.5.
Let µ λ,α,θ be the spectral measure of EHM: H λ,α,θ . For any α ∈ [0 , , there is afull measure set Θ = Θ( α ) ⊂ T such that for all θ ∈ Θ , the following hold: (1) For λ ∈ R , dim spe (cid:0) µ λ,α,θ (cid:1) = 1 if and only if β ( α ) = ∞ . (2) For λ ∈ R and for all α ∈ [0 , , dim spe (cid:0) µ λ,α,θ (cid:1) = 1 . (3) For λ ∈ R , dim spe (cid:0) µ λ,α,θ (cid:1) = 1 if β ( α ) > . We will see the explicit formula of the Lyapunov exponent and the spectral decomposition ofEHM, in section 7. In region R , EHM has positive Lyapunov for all α . Part (1) then followsfrom Theorem 2.3 directly. Region R is actually where EHM has purely absolutely continuousmeasure for all α and a.e. θ , see [3] and Theorem 7.2 in section 7. In view of Definition (2.3), itis well known that if a measure is absolutely continuous w.r.t. Lebesgue measure, then it has fullspectral dimension. Part (2) is then a direct consequence of a.c. spectrum and this fact. We listpart (2) here for completeness only. R is the region where EHM has zero Lyapunov exponent andpurely singular continuous spectrum for almost all ( θ, α ), part (3) follows from Theorem 2.1 andsome technical improvements of the (Λ , β ) bound for analytic sampling functions. We will discuss PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 7 in details about these three parts in section 7.Full spectral dimensionality is defined through the boundary behavior of the Borel transform ofthe spectral measure. It implies a range of properties, in particular, maximal packing dimensionand quasiballistic quantum dynamics. Thus our criterion links way a purely analytic property of thespectral measure to arithmetic property of the frequency in a sharp. In particular, consider H λ,α,θ ,the extended Harper’s model (EHM) given in (2.13). In this part, we will focus on EHM and discussthe consequences of the full spectral dimensionality in terms of these explicit parameters.Recall that the Hausdorff/packing dimension of a (Borel) measure µ , namely, dim H ( µ ) / dim P ( µ )is defined through the lim sup / lim inf ( µ almost everywhere) of its γ -derivativelim ε ↓ ln µ ( E − ε, E + ε )ln ε . If the lim inf is replaced by lim sup in the definition (2.3), we can define correspondingly the lowerspectral dimension dim spe ( µ ). It is well known (see e.g. [10, 14, 32]) the relation between thesefractal dimensions is dim H ( µ ) = dim spe ( µ ) ≤ dim spe ( µ ) ≤ dim P ( µ ). Therefore, lower bounds onspectral dimension lead to lower bounds on packing dimension, thus also for the packing/upper boxcounting dimensions of the spectrum as a set. We obtain corresponding non-trivial results for allthe above quantities. The lower bounds also provide explicit examples where the spectral measurehas different Hausdorff and packing dimension.Lower bounds on spectral dimension also have immediate applications to the lower bounds onquantum dynamics. Let δ j ∈ ℓ ( Z ) be the delta vector in the usual sense. For p >
0, define(2.15) h| X | pδ i ( T ) = 2 T Z ∞ e − t/T X n | n | p |h e − itH δ , δ n i| . The power law of h| X | pδ i ( T ) characterizes the propagation rate of e − itH δ . Define the upper/lowertransport exponents to be(2.16) β + δ ( p ) = lim sup T →∞ ln h| X | pδ i ( T ) p ln T , β − δ ( p ) = lim inf T →∞ ln h| X | pδ i ( T ) p ln T .β − δ ( p ) = 1 for all p > β + δ ( p ) = 1 for all p > β − δ ( p ) = 0 somtimes is called quasilocalized motion. It was proved in [19]that β + δ ( p ) ≥ dim P ( µ ) , ∀ p >
0. In view of Corollary 2.5, we have:
Corollary 2.6.
Let µ λ,α,θ be the spectral measure of EHM: H λ,α,θ defined in (2.13). For any α ∈ [0 , , there is a full measure set Θ = Θ( α ) ⊂ T such that for any θ ∈ Θ , H λ,α,θ has full packingdimension of µ λ,α,θ and quasiballistic motion if (1) λ ∈ R and β ( α ) = ∞ . (2) λ ∈ R and for all α ∈ [0 , . (3) λ ∈ R and β ( α ) > . Hausdorff dimension of the spectral measure is always equal to zero for a.e. phase for any ergodicoperator [39] in the regime of positive Lyapunov exponents. Combining the Lyapunov exponentformula of H λ,α,θ (see (7.3)) with the result of Simon in [39], we have dim H ( µ λ,α,θ ) = 0 for λ > θ and any α . In view of part (2) of Corollary 2.6, for λ >
1, a.e. θ and β ( α ) = ∞ , we have0 = dim H ( µ λ,α,θ ) < dim P ( µ λ,α,θ ) = 1 . In contrast to the Hausdorff dimension, the relation for the packing dimension only goes in one direction.
RUI HAN, FAN YANG AND SHIWEN ZHANG
We are also interested in the fractal dimensional properties of the density states measure andthe dimension of the spectrum as a set. Let d N λ,α be the density states measure and Σ λ,α be thespectrum of H λ,α,θ . For irrational α , they are both θ independent. It is well known thatd N λ,α = E θ ( µ λ,α,θ )(2.17)and Σ λ,α = supp top (d N λ,α ) . By these relations and the general properties of the packing dimensionof a measure and its topological support (see e.g. [14]), Corollary 2.6 implies thatdim P (d N λ,α ) = dim P (Σ λ,α ) = 1(2.18)in the corresponding parameter regions where EHM has full packing dimension.For the dynamical transport part, Last in [35] proved that almost Mathieu operator with anappropriate Liouville frequency has quasiballistic motion for the first time. In general, quasiballisticproperty is a G δ in any regular space, see e.g. [38, 17], thus this was known for (unspecified)topologically generic frequencies. In [32], the authors gave a precise arithmetic condition on α for thequasiballistic motion depending on whether or not Lyapunov exponent vanishes in the quasiperiodicSchr¨odinger setting. Here, we prodive the parametric conditions for the EHM. The conclusionscan also be extended directly to more general singular Jacobi operators with analytic quasiperiodicpotentials. 3. Preliminaries
We recall some commonly used notations for reader’s convenience. We denote L ∞ ( T , R ) and L ∞ ( T , C ) to be the space of all 1-periodic bounded functions, taking values in R and C respectively.Denote the usual L ∞ norm in both spaces by k f k ∞ := sup x ∈ T | f ( x ) | . Note we only require thediagonal potential function v to be real valued functions, all the other sampling function are allowedto take value in C . We do not emphasize the real/complex value anymore unless necessary. Denote L ( T , C ) to be the usual Lebesgue space with the 1-norm k f k := R T | f ( θ ) | d θ . Denote C ω ( T , C ) tobe the space of all 1-periodic analytic functions and denote C k ( T , C ) to be the space of all functionswith continuous k -th order derivatives for all k = 0 , , · · · , ∞ . We denote Lip( T , C ) to be the spaceof all 1-periodic Lipschitz continuous functions, induced with the Lipschitz norm given by: k f k Lip := k f k ∞ + sup x,y ∈ T | f ( x ) − f ( y ) || x − y | . (3.1)We identify the sequence u = { u n } n ∈ Z with u n whenever it is clear that n is the index. Denotethe ℓ ∞ norm of u ∈ ℓ ∞ ( Z , C ) by k u k ∞ := sup n ∈ Z | u n | . We will denote the distance on T by k θ k T := inf n ∈ Z | θ − n | and may drop the subindex k · k = k · k T whenever it is clear.3.1. Transfer matrices and Lyapunov exponents.
Let H be given as in (1.1):(3.2) ( Hu ) n = w n u n +1 + w n − u n − + v n u n , n ∈ Z . The eigenvalue equation Hu = Eu can be rewritten via the following skew product: (cid:18) u n +1 u n (cid:19) = A n ( E ) (cid:18) u n u n − (cid:19) , (3.3)where A n ( E ) = 1 w n D n ( E ) , D n ( E ) = (cid:18) E − v n − ¯ w n − w n (cid:19) . (3.4)For n ∈ N + and m ∈ Z , define the n-step transfer matrix at position m to be(3.5) A ( n, m ; E ) = n + m − Y j = m A j ( E ) , PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 9 (3.6) D ( n, m ; E ) = n + m − Y j = m D j ( E ) . We denote the scalar product of w n by the similar notation:(3.7) w ( n, m ) = n + m − Y j = m w j , m ∈ Z , n ∈ N + and denote A ( n ; E ) = A ( n, E ) , n > A (0; E ) = Id ; A ( n ; E ) = A − ( − n, n + 1; E ) , n < , (3.8) D ( n ; E ) = D ( n, E ) , n > D (0; E ) = Id ; D ( n ; E ) = D − ( − n, n + 1; E ) , n < , (3.9)(3.10) c ( n ) = c ( n, , n > , for simplicity.The (upper) Lyapunov exponent characterizes the grow(decay) rate of the norm of the transfermatrix k A ( n, m ) k , it will be convenient to introduce the Lyapunov exponent by using the dynamicalnotations. We refer readers to [32, 2] and references therein for the general definition of the Lyapunovexponent of linear skew product. In this part, we will restrict ourselves to the quasiperiodic cocycles.Let α ∈ R \ Q and A : T GL(2 , C ). We call ( α, A ) a (complex) cocycle. In view of (3.5) and (3.8),denote the transfer matrix in the quasiperiodic cocycle case by(3.11) A ( n ; θ, α ) = n Y j =1 A (cid:0) θ + ( j − α (cid:1) , θ ∈ T , n ∈ N + . The Lyapunov exponent is given by the formula:(3.12) L ( A, α ) = lim n → + ∞ n Z T ln k A ( n ; θ, α ) k d θ = inf n> n Z T ln k A ( n ; θ, α ) k d θ. For irrational α , the point-wise limit L ( A, α ) = lim n → + ∞ n ln k A ( n ; θ, α ) k also hold true for a.e. θ ∈ T by subadditive ergodic theory.By uniquely ergodicity of the irrational rotations we have the following uniform upper bound (in θ ) for both matrix and scalar cases: Lemma 3.1 (e.g. [15, 30]) . If A ∈ C ( T , GL(2 , C )) , then lim sup n → + ∞ n ln k A ( n ; θ, α ) k ≤ L ( A, α )(3.13) uniformly in θ ∈ T .If a ∈ C ( T , C ) and ln | a ( θ ) | ∈ L ( T ) , then lim sup n → + ∞ n ln | n Y j =1 a (cid:0) θ + ( j − α (cid:1) | ≤ Z T ln | a ( θ ) | d θ (3.14) uniformly in θ ∈ T . Remark . If a ∈ C ( T , C ) has no zeros, then a ( θ ) is also continuous. By (3.14), we have1 n ln (cid:12)(cid:12)(cid:12) n Y j =1 a (cid:0) θ + ( j − α (cid:1) (cid:12)(cid:12)(cid:12) ≤ Z T ln (cid:12)(cid:12)(cid:12) a ( θ ) (cid:12)(cid:12)(cid:12) d θ + ǫ (3.15) ⇐⇒ n Y j =1 | a (cid:0) θ + ( j − α (cid:1) | ≥ e n (cid:0) R T ln | a ( θ ) | d θ − ǫ (cid:1) (3.16)for n > n ( ǫ ) (uniform in θ ). This immediately gives the desired lower bound in (2.2) in a uniformway. If a ( θ ) has zeros, there is no such uniform lower bound for the scalar product anymore. Onetechnical achievement in the paper is, with some mild assumptions on the non-degeneracy of thezeros, we are able to get a weakened version of (3.15) (see Lemma 4.2), which will be sufficient forthe spectral continuity.3.2. The Weyl-Titchmarsh m -function and subordinacy theory. The boundary behavior ofthe m function is linked to the power law of the half line solution and the growth of the transfermatrix norm A ( n, m ; E ) via the well known Gilbert-Pearson subordinacy theory [16, 18]. We give abrief review on m -function and the subordinacy theory. More details can be found, e.g., in [7].Let H be as in (1.1) and z = E + iε ∈ C . Consider equation(3.17) Hu = zu. with the family of normalized phase boundary conditions:(3.18) u ϕ cos ϕ + u ϕ sin ϕ = 0 , − π/ < ϕ < π/ , | u ϕ | + | u ϕ | = 1 . Let Z + = { , , · · · } and Z − = {· · · , − , − , } . Denote by u ϕ = { u ϕj } j ≥ the right half linesolution on Z + of (3.17) with boundary condition (3.18) and by u ϕ, − = { u ϕ, − j } j ≤ the left halfline solution on Z − of the same equation. Also denote by v ϕ and v ϕ, − the right and left half linesolutions of (3.17) with the orthogonal boundary conditions to u ϕ and u ϕ, − , i.e., v ϕ = u ϕ + π/ , v ϕ, − = u ϕ + π/ , − . For any function u : Z + → C we denote by k u k ℓ the norm of u over a lattice interval oflength ℓ ; that is(3.19) k u k ℓ = h [ ℓ ] X n =1 | u n | + ( ℓ − [ ℓ ]) | u [ ℓ ]+1 | i / . Similarly, for u : Z − → C , we define(3.20) k u k ℓ = h [ ℓ ] − X n =1 | u − n | + ( ℓ − [ ℓ ]) | u − [ ℓ ] | i / . For any ε >
0, let ℓ = ℓ ( ϕ, ε, E ) be(3.21) k u ϕ k ℓ ( ϕ,ε ) k v ϕ k ℓ ( ϕ,ε ) = 12 ε .ℓ − ( ϕ ) is defined through the same equation by u ϕ, − , v ϕ, − . It is easy to check(3.22) k u ϕ k ℓ · k v ϕ k ℓ ≥
12 ([ ℓ ] − . Let m ϕ ( z ) : C + C + and m − ϕ ( z ) : C + C + the right and left Weyl-Titchmarsh m-functions(half line) associated with the boundary condition (3.18). Let m = m and m − = m − be the halfline m-functions corresponding to the Dirichlet boundary conditions. The following quantitativesubordinate theory was proved in [24], well known as Jitomirskaya-Last inequality. PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 11
Lemma 3.2 (Jitomirskaya-Last inequality, Theorem 1.1 in [24]) . For E ∈ R and ε > , the followinginequality holds for any ϕ ∈ ( − π , π ] : (3.23) 5 − √ | m ϕ ( E + iε ) | < k u ϕ k ℓ ( ϕ,ε ) k v ϕ k ℓ ( ϕ,ε ) < √ | m ϕ ( E + iε ) | . There is also one general statement about the existence of generalized eigenfunctions with sub-linear growth in its ℓ -norm: Lemma 3.3 ([36]) . For µ θ -a.e. E , there exists ϕ ∈ ( − π/ , π/ such that u ϕ and u ϕ, − both obey (3.24) lim sup ℓ →∞ k u k ℓ ℓ / ln ℓ < ∞ . This inequality provides us an upper bound for the ℓ -norm of the solution, which is crucial in theproof of the spectral singularity.The next proposition relates the whole line m-function M and half line m-function m ϕ , whichcan be found in [11]. Proposition 3.4 (Corollary 21 in [11]) . Fix E ∈ R and ε > , (3.25) | M ( E + iε ) | ≤ sup ϕ | m ϕ ( E + iε ) | . By this proposition, to bound M from above and get spectral continuity as in (2.4), it is enoughto obtain uniform upper bounds of m ϕ in boundary condition ϕ for the right half line problem.For spectral singularity, we need to consider both m ϕ ( z ) and m − ϕ ( z ). Let ( U ψ ) n = ψ − n +1 , n ∈ Z be a a unitary operator on ℓ ( Z ). Let e H = U HU − . Denote by e m, e m ϕ , e u ϕ and e ℓ ( ϕ ), correspondingly, m, m ϕ , u ϕ and ℓ ( ϕ ) of the operator e H . The following facts are well known in the past literatures(seee.g. section 3, [25]). For any ϕ ∈ ( − π/ , π/ M ( z ) = m ϕ ( z ) e m π/ − ϕ − m ϕ ( z ) + e m π/ − ϕ and(3.27) e ℓ ( π/ − ϕ ) = ℓ − ( ϕ ) , k u k ℓ = k U u k ℓ . In view of (2.10), a direct consequence of (3.26) is (e.g. Lemma 5 in [25]):
Lemma 3.5.
For any < γ < , suppose that there exists a ϕ ∈ ( − π/ , π/ such that for µ -a.e. E in some Borel set S , we have that lim inf ε → ε − γ | m ϕ ( E + iε ) | = ∞ and lim inf ε → ε − γ | e m π/ − ϕ ( E + iε ) | = ∞ . Then for µ -a.e. E in S , lim inf ε → ε − γ | M ( E + iε ) | = ∞ , namely, the restriction µ ( S ∩ · ) is γ -spectral singular. Continued fraction.
An important tool in the study of quasiperiodic sequence is the continuedfraction expansion of irrational numbers. Let α ∈ T \ Q , α has the following unique expression with a n ∈ N : α = 1 a + a + a ··· . (3.28)Let p n q n = 1 a + a + ··· + 1 an (3.29) be the continued fraction approximants of α . Let β ( α ) = lim sup n →∞ ln q n +1 q n .β ( α ) being large means α can be approximated very well by a sequence of rational numbers. Let usmention that { α : β ( α ) = 0 } is a full measure set.The following properties about continued fraction expansion are well-known:12 q n +1 ≤ k q n α k T ≤ q n +1 . (3.30)For any q n ≤ | k | < q n +1 , k q n α k T ≤ k kα k T . (3.31)Combining definition of β ( α ) (2.8) with (3.31), we have: If β ( α ) = 0, then for any δ >
0, for | k | large, the following inequality holds: k kα k T > e − δ | k | . (3.32)3.4. More about β -almost periodicity and the (Λ , β ) bound. In this paper, we considerbounded sequences v n , w n (for example, v n , w n are both generated by some smooth sampling func-tions). Let D ( n, m ) and w ( n, m ) be defined as in (3.6) and (3.7). The mild assumption on v n , w n yields the following trivial upper bound for D ( n, m ) and w ( n, m ): there is Λ = Λ ( k v k ∞ , k w k ∞ ) > n ∈ N , and any E ∈ N := N ( H ),sup m ∈ Z k D ( n, m ; E ) k ≤ e Λ n , (3.33) sup m ∈ Z | w ( n, m ) | ≤ e Λ n . (3.34)Suppose w n has (Λ , β )- q bound as in (2.2). Without of generality, we assume Λ = Λ for simplicity.For 1 ≤ r ≤ q and m ∈ Z , write w ( q, m ) = w ( q − r, m + r ) w ( r, m ). Combine (2.2) with the upperbound (3.34), we have min | m |≤ e δβ q | w ( r, m ) | ≥ e − q , ≤ r < q. (3.35)In particular, r = 1 gives min | m |≤ e δβ q | w m | ≥ e − q . (3.36)Assume further w n has β - q almost periodicity as in (2.1), by (3.36), β - q periodicity can be strength-ened as, max | m |≤ e δβ q (cid:12)(cid:12)(cid:12) w m ± q w m − (cid:12)(cid:12)(cid:12) < e − ( β − q (3.37)We also abuse the notation frequently by saying the operator H or the transfer matrix A ( n, E )has β -almost periodicity and (Λ , β ) boundedness if the corresponding v n , w n has β almost periodicityand (Λ , β ) boundedness .The lower bound on w ( n, m ) and upper bound on D ( n, m ) also imply that for any E ∈ N , and | m | ≤ e δβ q k A ( q, m ) k < e q , max ≤ r 4. (Λ , β ) bound for quasiperiodic smooth sequence and the proof of Corollary 2.2 Assume we have a the quasiperiodic sequence v ( θ + nα ) generated by a Lipschitz sampling function v . Let q n be given as in (2.8). By (2.8), for any 0 < β < β ( α ) / 2, there is a subsequence q n k suchthat ln q n k +1 > βq n k . Then for any θ, j and 1 ≤ n ≤ q n k , | v (cid:0) θ + mα (cid:1) − v (cid:0) θ + ( m ± q n k ) α (cid:1) | ≤ k v k Lip · k q n k α k ≤ k v k Lip · q n k +1 ≤ k v k Lip · e − βq nk ≤ e − βq nk , (4.1)provided q n k large. Same computation works for c . Therefore, v ( θ + nα ) and c ( θ + nα ) are β -almostperiodic for Lipschitz continuous v, c .The more challenging part is the (Λ , β ) bound on c ( θ + nα ), where we need some further assump-tion on c . We will focus on this throughout the rest of this section.The key ingredient for the proof of the (Λ , β ) bound is the following lemma in [1]: Lemma 4.1. Let α ∈ R \ Q , θ ∈ R and ≤ j ≤ q n − be such that | sin π ( θ + j α ) | = inf ≤ j ≤ q n − | sin π ( θ + jα ) | , then for some absolute constant C > , − C ln q n ≤ q n − X j =0 ,j = j ln | sin π ( θ + jα ) | +( q n − 1) ln 2 ≤ C ln q n . This lemma was used in [31] to prove some optimal singular continuous spectrum results. Byextending the argument in [31] to exponentially many periods, we are able to prove (Λ , β ) bound forany analytic sampling function. Actually, we can deal with more general sampling functions withmuch weaker regularities. Define F ( T , C ) := n c ∈ L ∞ ( T , C ) : ∃ m ∈ N + , θ ℓ ∈ T , τ ℓ ∈ (0 , , ℓ = 1 , · · · , m such that g ( θ ) := c ( θ ) Q mℓ =1 | sin π ( θ − θ ℓ ) | τ ℓ ∈ L ∞ ( T , C ) and inf T | g ( θ ) | > . o (4.2)Suppose c ( θ ) ∈ F ( T , C ) with θ ℓ and g ( θ ) given as in (4.2) such that c ( θ ) = g ( θ ) m Y ℓ =1 | sin π ( θ − θ ℓ ) | τ ℓ . (4.3)Clearly, ln | g ( θ ) | ∈ L ( T ). By the well known integral R T ln | sin πθ | d θ = − ln 2, it is easy to checkthat ln | c ( θ ) | ∈ L ( T ) and is linked to ln | g | by: Z T ln | c ( θ ) | d θ = Z T ln | g ( θ ) | d θ − ln 2 m X ℓ =1 τ ℓ . (4.4)The following technique lemma shows that any sampling function in F ( T , C ) with an irrationalforce can generate a (Λ , β ) bounded sequence. Lemma 4.2. Assume that there exists m ∈ N + , θ ℓ ∈ T , τ ℓ ∈ (0 , , ℓ = 1 , · · · , m, g ( θ ) ∈ L ∞ ( T , C ) such that inf T | g ( θ ) | > and c ( θ ) = g ( θ ) m Y ℓ =1 | sin π ( θ − θ ℓ ) | τ ℓ . (4.5) Then for any α with < β < β ( α ) and < δ < P mℓ =1 τ ℓ P mℓ =1 τ ℓ , there is a sequence q n → ∞ and a fullLebesgue measure set Θ = Θ( α, θ , · · · , θ m ) such that for any θ ∈ Θ and q n large enough , c ( θ + nα ) satisfies: min | k |≤ q − n e δβ qn ( k +1) q n − Y j = kq n | c ( θ + jα ) | > e − Λ q n , (4.6) where Λ := Λ ( τ, g, δβ ) = ln 2 m X ℓ =1 τ ℓ − ln (cid:0) inf T | g ( θ ) | (cid:1) + δ min { β, } . (4.7) Assume further g ( θ ) ∈ C ( T , C ) and ln | g ( θ ) | ∈ L ( T ) , Λ in (4.7) can be replaced by Λ = − Z T ln | c ( θ ) | d θ + 2 δ min { β, } . (4.8) Moreover, c ( θ + nα ) is (Λ , β ) bounded as defined in (2.2) such that min | m |≤ e δβ qn m + q n − Y j = m | c ( θ + jα ) | > e − Λ q n , θ ∈ Θ(4.9) where Λ = − Z T ln | c ( θ ) | d θ + 6 δ min { β, } . (4.10) Remark . The above Λ and Λ can be negative in general, which makes (4.6 and (4.9) actuallyexponentially grow (instead of decay). This is natural since there is actually a ‘large’ scaling of size R ln | g | ∼ R ln | c | for the product in these cases. We are more interested in the case where R ln | c | ≤ and Λ are indeed positive. In particular, it is always possible to re-scale c ( θ ) to make thelogarithm average zero. This will lead to an arbitrarily small Λ (positive) in (4.10). Combine thiswith the uniform Lyapunov upper bound for (3.33) ([15]), we can get some refined results in the zeroLyapunov regime, e.g., the critical EHM model, about the spectral continuity and quasi-ballisticmotion. See more discussion in the next Corollary 4.3 and section 7 about Corollary 2.5. Remark . It is an easy exercise that if c ( θ ) is C k continuous on T with finitely many zeros withnon-degenerate k -th order derivatives, then c ( θ ) ∈ F ( T , C ) T Lip( T , C ) with a continuous g ( θ ) for all k ≥ 1. By Lemma 4.2, there is Λ = Λ( c ) such that c ( θ + nα ) is (Λ , β ) bounded for any 0 < β < β ( α )and a.e. θ ∈ T . From the proof of Lemma 4.2, see (4.11), the subsequence q n k can be taken to besame as in the β -almost periodicity (4.1). Therefore, Corollary 2.2 follows from Theorem 2.1. Weomit the details here. An interesting question is whether the non-degenerate condition on c can beweakened and what is the appropriate ‘non-degenerate’ condition on any Lipschitz function suchthat (4.2) holds. Proof. Let 0 < β < β ( α ) and q n be defined as in (2.8). For any δ > 0, let q n k be the subsequencesuch that ln q n k +1 ≥ β q n k . For simplicity, drop the subindex n k and denote the subsequence stillby q n , i.e., q n +1 > e β q n (4.11) The sequence itself only depends β ( α ), while the largeness depends on θ, α, β, δ, τ . PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 15 We also write ˜ β = min { β, } . It is obvious that for any m ∈ Z and θ ∈ T , ( k +1) q n − Y j = kq n | g ( θ + jα ) | > (cid:0) inf T | g ( θ ) | (cid:1) q n = e q n ln(inf T | g ( θ ) | ) . (4.12)In view of (4.5), it is enough to study the lower bound for each k θ − θ ℓ k T . For any α ∈ [0 , \ Q andany θ ℓ , ℓ = 1 , · · · , m , letΘ ℓ := [ γ> n θ ∈ T : k θ − θ ℓ + nα k T ≥ γ | n | − , ∀ n ∈ Z \{ } o (4.13)It is well known that Θ ℓ is a full measure set. LetΘ := m \ ℓ =1 Θ ℓ . (4.14)For any θ ∈ Θ and 1 ≤ ℓ ≤ m , there is γ ℓ = γ ( θ, θ ℓ , α ) > k θ − θ ℓ + nα k T ≥ γ ℓ | n | , ∀ n ∈ Z \{ } . (4.15)For all 1 ≤ ℓ ≤ m and | k | < q − n e δβq n , let j ℓ,k ∈ [0 , q n ) be such that the following holds: | sin π (cid:0) θ − θ ℓ + kq n α + j ℓ,m α (cid:1) | = inf ≤ j 1) ln 2 − C ln q n ≥ e − q n ln 2 − τ − δ e βq n (4.18)provided C ln q n < τ − δ e βq n where C is the absolute constant in Lemma 4.1 and τ is the same in(4.17). Now putting (4.17) and (4.18) together, we have ( k +1) q n − Y j = kq n (cid:16) m Y ℓ =1 | sin π ( θ + jα − θ ℓ ) | τ ℓ (cid:17) = m Y ℓ =1 (cid:0) q n − Y j =0 | sin π ( θ − θ ℓ + kq n α + jα ) | (cid:1) τ ℓ = (cid:16) m Y ℓ =1 (cid:0) q n − Y j =0 ,j = j l,k | sin π ( θ − θ ℓ + kq n α + jα ) | (cid:1) τ ℓ (cid:17) · (cid:16) m Y ℓ =1 | sin π ( θ − θ ℓ + kq n α + j l,k α ) | τ ℓ (cid:17) ≥ (cid:16) m Y ℓ =1 (cid:0) e − q n ln 2 − τ − δ e βq n (cid:1) τ ℓ (cid:17) · (cid:16) m Y ℓ =1 (cid:0) e − τ − δ e βq n (cid:1) τ ℓ (cid:17) = e − q n (ln 2 P mℓ =1 τ ℓ ) − δ e βq n . Combined with (4.12), we have that for all | k | ≤ q − n e δβ q n , ( k +1) q n − Y j = kq n | c ( θ + jα ) | > e − (cid:0) ln 2 P mℓ =1 τ ℓ − ln inf T | g ( θ ) | + δ e β (cid:1) q n (4.19)provided q n > ˜ q = ˜ q (cid:0) max ℓ γ − ℓ , δ, α, P mℓ =1 τ ℓ (cid:1) .Assume further g ( θ ) , c ( θ ) ∈ C ( T , C ). Since inf | g ( θ ) | > 0, ln | g ( θ ) | − is also continuous. ByLemma 3.1, there is n = n ( δ e β ) such that the following upper bound holds uniform in θ ∈ T for n > n : 1 n n X j =1 ln | g ( θ + jα ) | − ≤ Z T ln | g ( θ ) | − d θ + δ e β. (4.20)In particular, for all q n ≥ n and any k ∈ Z we have (cid:16) q n − Y j =0 | g ( θ + kq n α + jα ) | − (cid:17) qn ≤ e − R T ln | g ( θ ) | d θ + δ e β = ⇒ ( k +1) q n − Y j = kq n | g ( θ + jα ) | ≥ e q n (cid:0) R T ln | g ( θ ) | d θ − δ e β (cid:1) . Therefore, we can replace ln (cid:0) inf T | g ( θ ) | (cid:1) in (4.19) by R T ln | g ( θ ) | d θ − δ min { β, } . In view of(4.4), we haveΛ = ln 2 m X ℓ =1 τ ℓ − Z T ln | g ( θ ) | d θ + 2 δ min { β, } = − Z T ln | c ( θ ) | d θ + 2 δ min { β, } , (4.21)which gives the desired expression of Λ in (4.8).Let e c ( θ ) = c ( θ ) e − R T ln | c ( θ ) | d θ . It is easy to check that R T ln | e c ( θ ) | d θ = 0. By (4.21), we have forall | k | ≤ q − n e δβ q n , ( k +1) q n − Y j = kq n | e c ( θ + jα ) | > e − δ e β q n . (4.22) PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 17 By Lemma 3.1, there is r = r ( δ e β ) ∈ N such that for any m ∈ Z and r ≥ r , m + r − Y j = m | e c ( θ + jα ) | ≤ e r (cid:0) R T ln e c ( θ ) | d θ + δ e β (cid:1) = e δ e β r . (4.23)For 0 ≤ r < r , we have the trivial upper bound Q m + r − j = m | e c ( θ + jα ) | ≤ e r ln( k e c k ∞ ) ≤ e r ln( k e c k ∞ +1) .Therefore, for any m ∈ Z and 1 ≤ r ≤ q n , m + r − Y j = m | e c ( θ + jα ) | ≤ e δ e β q n (4.24)provided q n ≥ δ − e β − r ln( k e c k ∞ + 1).Then for any | m | < e δβ q n , there is k such that kq n ∈ ( m, m + q n ], therefore, m + q n − Y j = m | e c ( θ + jα ) | = kq n − Y j = m | e c ( θ + jα ) | · m + q n − Y j = kq n | e c ( θ + jα ) | (4.25) = Q kq n − j =( k − q n | e c ( θ + jα ) | Q m − j =( k − q n | e c ( θ + jα ) | · Q ( k +1) q n − j = kq n | e c ( θ + jα ) | Q ( k +1) q n − j = m + q n | e c ( θ + jα ) | (4.26) > e − δ e β q n e δ e β q n · e − δ e β q n e δ e β q n (4.27) = e − δ e β q n . (4.28)Therefore, m + q n − Y j = m | c ( θ + jα ) | = e q n R T ln | c ( θ ) | d θ m + q n − Y j = m | e c ( θ + jα ) |≥ e − ( − R T ln | c ( θ ) | d θ +6 δ e β ) q n =: e − Λ q n , (4.29)as claimed. This completes the proof of Lemma 4.2. (cid:3) As an explicit example, we have the following arbitrarily slow lower bound for the analytic casewith zero ln mean. Corollary 4.3. Assume that c ( θ ) ∈ C ω ( T , C ) and R T ln | c ( θ ) | d θ = 0 . Denote the all zeros of c ( θ ) on T by c − (0) = { θ , · · · , θ m } . For any β with < β < β ( α ) and < δ < , there is there isa sequence q n → ∞ and a full Lebesgue measure set Θ = Θ( α, c − (0)) such that for any θ ∈ Θ , c ( θ + nα ) satisfies: min | k |≤ q − n e δβ qn ( k +1) q n − Y j = kq n | c ( θ + jα ) | > e − δ min { β, } q n , (4.30) min | m |≤ e δβ qn m + q n − Y j = m | c ( θ + jα ) | > e − δ min { β, } q n . (4.31) Clearly, analytic function c ( θ ) only has finitely many zeros on T . Proof. Clearly, there is an analytic function e g ( θ ) such that: c ( θ ) = e g ( θ ) m Y ℓ =1 (cid:0) e π i θ − e π i θ ℓ (cid:1) , inf T | e g ( θ ) | > . (4.32)Direct computation shows R T ln | e g ( θ ) | d θ = R T ln | c ( θ ) | d θ = 0 and c ( θ ) = e g ( θ ) m Y ℓ =1 (cid:0) e π i θ − e π i θ ℓ (cid:1) = e g ( θ )(2i) m m Y ℓ =1 e i π ( θ + θ ℓ ) sin π ( θ − θ ℓ ) . (4.33)Therefore, | c ( θ ) | Q mℓ =1 | sin π ( θ − θ ℓ ) | = 2 m | e g ( θ ) | . (4.34)Apply Lemma 4.2 to (4.33) where τ = · · · = τ m = 1 and | g ( θ ) | ≡ m | e g ( θ ) | , we have (4.6) and(4.9) hold with Λ = 2 δ min { β, } and Λ = 6 δ min { β, } . (cid:3) Spectral continuity: proof of Theorem 2.1 Following the notations and assumptions in Theorem 2.1, consider(5.1) ( Hu ) n = w n u n +1 + w n − u n − + v n u n , n ∈ Z . Assume that there are positive constants β, δ, Λ > q n → ∞ such that w n , v n has β - q n almost periodicity and w n has (Λ , β )- q n bound.The key observation is: if H has β - q n almost periodicity, then it can be approximated by a q n periodic operator exponentially fast in a finite (exponentially large) lattice. The estimates on the q n periodic operator eventually lead to the quantitative upper bound for the m -function as in (2.4)through the subordinacy theory Lemma 3.2.In view of Lemma 3.2, let v ϕ be the right half line solution to Hu = Eu with initial condition ϕ and ℓ = ℓ ( ϕ, ε, E ) is defined as in (3.21). As a direct consequence of Lemma 3.2 and Proposition3.4, the following relation between the power law of k v ϕ k ℓ and the spectral continuity was provedin [32] (see Lemma 2.1 and the proof of Theorem 6 there). Lemma 5.1. Fix < γ < . Suppose for µ -a.e. E , there is a sequence of positive numbers η k → and L k = ℓ k ( ϕ, η k , E ) → ∞ such that for any ϕ (5.2) 1 / (cid:0) L k (cid:1) γ ≤ k v ϕ k L k ≤ (cid:0) L k (cid:1) − γ . Then the spectral measure µ is γ -spectral continuous. Let A ( n ; E ) be defined as in (3.8). Denote by Tr A the trace of any matrix A ∈ GL(2 , C ). Thefollowing estimate on Tr A ( q n ; E ) is the key to prove the above power law and spectral continuity. Theorem 5.2. Let H, β, δ, Λ and q n be given as in (5.1). Suppose β > δ )Λ , then for µ a.e. E , there exists K ( E ) ∈ N , for k ≥ K ( E ) , we have | T rA ( q k ; E ) | < − e − q k . (5.3) For any < γ < , assume further that β > δ ) Λ1 − γ , (5.4) we have the power law required by (5.2). PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 19 Let C = C ( δ, Λ) = 300(1 + 1 δ )Λ . (5.5)Combining Lemma 5.1 and (5.4) in Theorem 5.2, if β > C , then µ is γ -spectral continuous for any γ < − Cβ < spe ( µ ) ≥ − Cβ . This proves Theorem 2.1.The trace estimate (5.3) shows that spectrally almost everywhere, A ( q k ; E ) is strictly ellipticeventually. The quantitative estimate (5.3) allows us to iterate the transfer matrix up to the lengthscale e Λ q k , which gives a well control on the norm of A ( q k ; E ). The norm estimate eventually leadsto the power law as required in (5.2) through (3.3).The proof of (5.4) and the required power law follows the outline of the Schr¨odinger case (see [32],Lemma 2.1). The main difference is now the transfer matrix A ( n ; E ) is in GL (2 , C ). We need toconsider some transformations introduced in [22] which conjugate A ( n ; E ) to some SL(2 , R ) matrix.Then many important techniques developed in [32] for SL(2 , R ) cocycles are now applicable. Thetrace estimate (5.3) leads to a norm estimate of A ( q k ; E ) and eventually leads to the estimate (5.2)for the truncated ℓ norm of the eigenfunction v ϕ by (3.3). We will omit the details here and focuson the proof of the trace estimate (5.3). For the sake of completeness, we sketch the proof of (5.4)and the power law (5.2) in the Appendix A.1 for reader’s convenience.The rest of the section is organized as follows: In section 5.1, we introduce the transformation wewill use to conjugate GL(2 , C ) to SL(2 , R ) and develop all the useful lemmas about the conjugate.In section 5.2, we study the case where the trace of the transfer matrix is greater than 2. In section5.3, we study the case where the trace of the transfer matrix is close to 2.Throughout this section, we assume v n , w n have β - q almost periodicity and w n has (Λ , β )- q boundfor some q large enough such that e − ( β − q < / 10. We also use the induced estimates (3.35)-(3.38)discussed in section 3.4 directly, refered also as β - q almost periodicity and (Λ , β )- q bound.5.1. Conjugate between SL(2 , R ) and GL (2 , C ) matrices. The trace estimate (5.3) was firstproved in [32] for SL(2 , R ) cocycles. The generalization to GL (2 , C ) case is very delicate. We needto consider the following transformation: let T n = q w n w n ! (5.6)and r n = w n +1 p | w n +1 w n | . (5.7)Let A n ( E ) be given as in (3.4). Define e A n ( E ) := r n − T − n A n T n − = 1 p | w n w n − | (cid:18) E − v n −| w n − || w n | (cid:19) . (5.8)The n-transfer matrix e A ( n, m ; E ) and e A ( n ; E ) for e A n will be defined in the same way as in (3.5):(5.9) e A ( n, m ; E ) = n + m − Y j = m e A j ( E ) , n ∈ N + , m ∈ Z and(5.10) e A ( n ; E ) = e A ( n, E ) , n > e A (0; E ) = Id ; e A ( n ; E ) = e A − ( − n, n + 1; E ) , n < . We also denote the scalar product of r n in the same way as w ( n, m ) in (3.10) for n ∈ N + ,(5.11) r ( n, m ) = n + m − Y j = m r j , m ∈ Z . Direct computation shows e A ( n, m ; E ) = r ( n, m − T − n + m − A ( n, m ; E ) T m − . (5.12)In view of (5.6) and (5.8), it is easy to check that k T n k = 1 and e A n , e A ( n, m ) ∈ SL(2 , R ) for any n, m . By (5.8) and (5.12), we are able to apply the techniques developed in [32] for SL(2 , R ) matrixand then swtich between the singular GL(2 , C ) case and the SL(2 , R ) case.The β -almost periodicity and the (Λ , β ) boundedness of w n imply the β -almost periodicity of r and T in the following sense: Lemma 5.3. If β > , then for all m ∈ Z such that | m | < e δβq , (cid:12)(cid:12)(cid:12) | r ± ( q, m ) | − (cid:12)(cid:12)(cid:12) < e − ( β − q (5.13) k T − m + q · T m − I k = k T m · T − m + q − I k < e − ( β − q . (5.14) Assume further that N ∈ N + , N q ≤ e δβq , then (cid:12)(cid:12)(cid:12) | r ± ( N q, | − (cid:12)(cid:12)(cid:12) < N e − ( β − q (5.15) k T · T − Nq − I k = k T − Nq · T − I k < N e − ( β − q . (5.16)Note that r ( n, m ) and T n are essentially scalar products, the proof is based on the following directcomputation: Proof. Set z m = w m w m + q . By (3.37), for | m | ≤ e δβq and q large, (cid:12)(cid:12) | z m | ± − (cid:12)(cid:12) ≤ | z ± m − | < e − ( β − q < . Clearly, | r n | = q | w n +1 || w n | . In view of (5.11) and (5.6), we have (cid:12)(cid:12)(cid:12) | r ( q, m ) | − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)s | c m +1 || c m | · · · | c m + q || c m + q − | − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)p | z m | − − (cid:12)(cid:12)(cid:12) < e − ( β − q (5.17)and T − m + q T m = q w m + q w − m + q ! (cid:18) p w m w − m (cid:19) = w m w − m + q | w m w − m + q | ! = (cid:18) z m | z m | (cid:19) Therefore, k T − m + q T m − I k ≤ (cid:12)(cid:12)(cid:12) z m | z m | − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) z m − | z m | (cid:12)(cid:12)(cid:12) | z m | ≤ (cid:12)(cid:12) z m − (cid:12)(cid:12) | z m | ≤ e − ( β − q . In particular, in (5.17), let m = 0 , q, q, · · · , ( N − q for N q < e δβq . Direct computation showsthat (cid:12)(cid:12)(cid:12) | r ( N q, | − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N − Y k =0 | r ( q, kq ) | − (cid:12)(cid:12)(cid:12) < N e − ( β − q , PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 21 T − Nq T = q w Nq w − Nq ! q w w − ! = w w − Nq | w w − Nq | ! , and k T − Nq T − I k ≤ (cid:12)(cid:12)(cid:12) w w − Nq | w w − Nq | − (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) N − Y k =0 z kq | z kq | − (cid:12)(cid:12)(cid:12) ≤ N − X k =0 (cid:12)(cid:12)(cid:12) z kq | z kq | − (cid:12)(cid:12)(cid:12) ≤ N e − ( β − q . (cid:3) (5.12) only implies k e A ( q ; E ) k ≈ k A ( q ; E ) k , while k A ( q ; E ) − e A ( q ; E ) k is not necessarily small.Lemma 5.3 actually shows e A ( q ; E ) and A ( q ; E ) are close to each other up to a conjugate. This willbe enough to control the difference between their traces. Fix E , we write A ( n, m ) = A ( n, m ; E ) forshort. Lemma 5.4. For all m ∈ Z such that | m | < e δβq , let Φ = Arg r ( q, m ) be the Principal value of r ( q, m ) ∈ C . For β > , k A ( q, m ) k ≤ k e A ( q, m ) k ≤ k A ( q, m ) k < e q (5.18) k e A ( q, m + 1) − e iΦ T − m A ( q, m + 1) T m k < e − ( β − q (5.19) and consequently, (cid:12)(cid:12)(cid:12) | T r e A ( q, m ) | − | T rA ( q, m ) | (cid:12)(cid:12)(cid:12) < e − ( β − q . (5.20) Proof. By (5.13), we have | r ± ( q, m ) | ≤ 2. (5.18) follows from (3.38) and (5.12) since k T ± m k = 1.By (5.12), we have e A ( q, m + 1) = r ( q, m ) T − m + q A ( q, m + 1) T m = (cid:0) | r ( q, m ) | T − m + q T m (cid:1) e iΦ T − m A ( q, m + 1) T m (5.21)Therefore, k e A ( q, m + 1) − e iΦ T − m A ( q, m + 1) T m k = k (cid:0) | r ( q, m ) | T − m + q T m − I (cid:1) e iΦ T − m A ( q, m + 1) T m k (5.22) ≤ k (cid:0) | r ( q, m ) | T − m + q T m − I (cid:1) k · k e iΦ T − m A ( q, m + 1) T m k (5.23) ≤ e − ( β − q k A ( q, m + 1) k (5.24) ≤ e − ( β − q . (5.25)The last inequality follows from (5.13) and (5.14) since k (cid:0) | r ( q, m ) | T − m + q T m − I (cid:1) k ≤ (cid:12)(cid:12) | r ( q, m ) | − (cid:12)(cid:12) · k T − m + q T m − I k + (cid:12)(cid:12) | r ( q, m ) | − (cid:12)(cid:12) + k T − m + q T m − I k . (5.20) follows directly from (5.19) since | Tr A ( q, m + 1) | = (cid:12)(cid:12)(cid:12) Tr (cid:0) e iΦ T − m A ( q, m + 1) T m (cid:1)(cid:12)(cid:12)(cid:12) . (cid:3) Standard telescoping argument allows us to pass the β -almost periodicity from the sequences w n , v n to the matrices A ( n, m ) , e A ( n, m ), up to product length q , Lemma 5.5. For all m ∈ Z such that | m | < e δβq , β > , k A ( q, m ; E ) − A ( q, m + q ; E ) k ≤ e ( − β +6Λ) q (5.26) and k e A ( q, m ; E ) − e A ( q, m + q ; E ) k ≤ e ( − β +6Λ) q . (5.27) Proof. Write m ′ = m + q for short. A ( q, m ) − A ( q, m ′ ) = q − X j =0 A ( q − j − , m + j + 1) (cid:16) A m + j − A m ′ + j (cid:17) A ( j, m ′ )= q − X j =0 D ( q − j − , m + j + 1) w ( q − j − , m + j + 1) (cid:16) D m + j w m + j − D m ′ + j w m ′ + j (cid:17) D ( j, m ′ ) w ( j, m ′ ) . By the trivial upper bound (3.33) for D ( n, m ) and the lower bound (3.35) for w ( n, m ), we have k A ( q, m ) − A ( q, m ′ ) k ≤ q − X j =0 e ( q − j − | w ( q − j − , m + j ) | (cid:12)(cid:12)(cid:12) w m ′ + j D m + j − w m + j D m ′ + j (cid:12)(cid:12)(cid:12) e Λ j | w ( j + 1 , m ′ ) |≤ q − X j =0 e ( q − j − e − q (cid:12)(cid:12)(cid:12) w m ′ + j D m + j − w m + j D m ′ + j (cid:12)(cid:12)(cid:12) e j Λ e − q ≤ q e q max | m |≤ e δβq (cid:12)(cid:12)(cid:12) w m + q D m − w m D m + q (cid:12)(cid:12)(cid:12) ≤ q e q max | m |≤ e δβq (cid:16)(cid:12)(cid:12) ( w m + q − w m ) D m (cid:12)(cid:12) + (cid:12)(cid:12) w m ( D m + q − D m ) | (cid:17) ≤ q e q e Λ e − βq ≤ e − ( β − q provided sup n | w n | , sup n,E k D n k ≤ e Λ and q large such that 2 qe Λ ≤ e Λ q .Let e w n = p | w n w n − | , e D n = (cid:18) E − v n −| w n − || w n | (cid:19) . Define e w ( n, m ) , e D ( n, m ; E ) exact in the sameas for w, D . It is easy to check that the Λ bounds of w n and D n hold true for e w n , e D n : | e w ( n, m ) | ≤ e Λ n , sup E k e D ( n, m ; E ) k ≤ e Λ n , ∀ n ≥ , m ∈ Z (5.28)and | ˜ w ( r, m ) | ≥ e − q , ≤ r ≤ q, | m | ≤ e δβq . (5.29)The β -almost periodicity of w n , v n are also passed directly to e w n , e D n :max | m |≤ e δβq | e w m − ˜ w m ± q | ≤ e Λ e − βq , max | m |≤ e δβq k e D m − e D m ± q k ≤ e − βq . (5.30)By the definition of e A n in (5.8), we have e A n = e w n e D n . Exact the same computation proves that k e A ( q, m ) − e A ( q, m ′ ) k ≤ q (2 e + e Λ ) e q e − βq ≤ e − ( β − q . (5.31) (cid:3) PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 23 The above telescoping argument can not be extended to exponential scale e Λ q as for r and T in(5.15),(5.16) directly. One main reason is we lose control of the matrix norm super-exponentially as k D ( e Λ q ) k . e Λ e Λ q . Such gowth can not be controlled by condition such as β & Λ. The key to provethe trace estimate (5.3) is to avoid using such rough bound for the matrix norm at an exponentialscale. This is one breakthrough in [32]. By all the above estimates of the conjugate r, T and somesimple linear algebra facts of SL(2 , R ) matrix found in [32], we are able to prove this extension forthe GL(2 , C ) case. We will see more details in the next two subsections.Similar to [32], we consider the following two cases where | T rA ( q ) | is away from 2 and close to 2.5.2. The case where the trace is away from 2. We start with the hyperbolic case in thefollowing sense: let S q = { E : | Tr A ( q ; E ) | > e − q } (5.32)We may fix E and write A ( q ) = A ( q ; E ) for simplicity whenever it is clear. Lemma 5.6. Let q n be given as in Theorem 5.2. If β > (260 + δ )Λ , then the set lim sup n →∞ S q n = n E : E belongs to infinitely many S q n o (5.33) has spectral measure zero. Lemma 5.4 implies for large β , Tr e A ( q ; E ) and Tr A ( q ; E ) lie in the same region, i.e., if E ∈ S q ,then | Tr e A ( q ; E ) | > e − q − e − ( β − q > e − q , (5.34)provided e ( β − q > 12 .The following linear algebra facts were proved in [32] Lemma 5.7. Suppose G ∈ SL(2 , R ) with < | Tr G | ≤ . The invertible matrix B such that (5.35) G = B (cid:18) ρ ρ − (cid:19) B − where ρ ± are the two conjugate real eigenvalues of G with | det B | = 1 satisfies (5.36) k B k = k B − k < p k G k p | Tr G | − If | Tr G | > , then k B k ≤ √ k G k √ | Tr G |− . Apply the above lemma to e A ( q ; E ) ∈ SL(2 , R ) satisfying (5.18) and (5.34),we have the followingdecomposition(5.37) e A ( q ) = B (cid:18) ρ ρ − (cid:19) B − where ρ ± are the two conjugate real eigenvalues of e A ( q ) with | ρ | > | Tr e A ( q ) | − > e − q and B satisfies | det B | = 1 and(5.38) k B k = k B − k < e q . By (5.37) and (5.38), we have that for any N ∈ N + , e A N ( q ) := [ e A ( q )] N = B (cid:18) ρ N ρ − N (cid:19) B − , k e A N ( q ) k ≤ e q | ρ | N (5.39) In the rest of this section, consider β > δ Λ and set N = [ e q ] < e δβq . (5.40)The above decomposition now turns the matrix product [ e A ( q )] N into a scalar product of ρ N with auniformly controlled conjugate B (independent of N ). This is one key algebra ingredient observedin [32]. This technique now allows us to extend the orbit of e A ( q ) to the exponentially long scale N = e q .The following technique lemma was proved in [32] (see Lemma A.1 there): Lemma 5.8. Suppose G is a two by two matrix satisfying (5.41) k G j k ≤ M < ∞ , for all < j ≤ N ∈ N + , where M ≥ only depends on N . Let G j = G + ∆ j , j = 1 , · · · , N, be a sequence of two by twomatrices with (5.42) δ = max ≤ j ≤ N k ∆ j k . If (5.43) N M δ < / , then for any ≤ n ≤ N (5.44) k n a j =1 G j − G n k ≤ N M δ. Let N = [ e q ], G = ρ e A ( q ) and G j = ρ e A ( q, jq + 1), | j | = 0 , , · · · N . By (5.18) and (5.27), it iseasy to check that k G j k ≤ e q and k G j − G k ≤ N e ( − β +6Λ) q ≤ e ( − β +67Λ) q . The above lemma isapplicable provided β > (260 + δ )Λ. One can prove that k e A ( N q ) − e A N ( q ) k ≤ | ρ | N e ( − β +260Λ) q (5.45) k e A ( − N q ) − e A − ( N q ) k ≤ | ρ | N e ( − β +260Λ) q (5.46)The proof (5.45) and (5.46) is a direct application of Lemma 5.8 and resembles the proof of Claim3, [32]. We omit the details here.Similar to (5.19), we can prove A ( ± N q ) and e A ( ± N q ) are close to each other up the size | ρ | N . Lemma 5.9. Let η = r − ( N q, , ζ = r ( N q, − N q ) and φ = Arg η, ψ = Arg ζ be the Principal valuesof η and ζ accordingly. For β > (260 + δ )Λ , k A ± ( N q ) − e ± i φ T e A ± ( N q ) T − k < e ( − β +127Λ) q | ρ | N , (5.47) k A ( − N q ) − e i ψ T e A ( − N q ) T − k < e ( − β +127Λ) q | ρ | N , (5.48) and consequently, k A − ( N q ) − e − i( φ + ψ ) A ( − N q ) k < e ( − β +260Λ) q | ρ | N . (5.49) Proof. By (5.8), A ( N q ) = η T Nq e A ( N q ) T − = (cid:0) | η | T Nq T − (cid:1) e i φ T e A ( N q ) T − . (5.50) PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 25 Therefore, by (5.15) and (5.16), k A ( N q ) − e i φ T e A ( N q ) T − k ≤ (cid:13)(cid:13) | η | T Nq T − − I (cid:13)(cid:13) · k e i φ T e A ( N q ) T − k≤ e ( − β +63Λ) q k e A ( N q ) k≤ e ( − β +127Λ) q | ρ | N , provided e ( β − q > e Λ q > 12. The last inequality follows from (5.39) and (5.45): k e A ( N q ) k ≤ k e A N ( q ) k + e ( − β +260Λ) q | ρ | N ≤ e q | ρ | N + e ( − β +260Λ) q | ρ | N ≤ e q | ρ | N . Note that e A ( N q ) ∈ SL(2 , R ), then k e A − ( N q ) k = k e A ( N q ) k ≤ e q | ρ | N . The proof for A − ( N q )is exactly the same since A − ( N q ) = η − T e A − ( N q ) T − Nq = e − i φ T e A − ( N q ) T − (cid:0) | η | − T T − Nq (cid:1) . (5.51)(5.8) and (5.10) imply that A ( − N q ) = A − ( N q, − N q + 1) = h r − ( N q, − N q ) T e A ( N q, − N q + 1) T − − Nq i − = r ( N q, − N q ) T − Nq e A ( − N q ) T − . (5.46) implies that k e A ( − N q ) k ≤ k e A − ( N q ) k + 2 | ρ | N e ( − β +260Λ) q ≤ e q | ρ | N . Now by (5.15)and (5.16), exact the same argument for (5.47) proves (5.48) provided e ( β − q > e Λ q > k A − ( N q ) − e − i( φ + ψ ) A ( − N q ) k ≤k A − ( N q ) − e − i φ T e A − ( N q ) T − k + k e − i φ T e A − ( N q ) T − − e − i φ T e A ( − N q ) T − k + k e − i φ T e A ( − N q ) T − − e − i( φ + ψ ) A ( − N q ) k≤k A − ( N q ) − e − i φ T e A − ( N q ) T − k + k e A − ( N q ) − e A ( − N q ) k + k e i ψ T e A ( − N q ) T − − A ( − N q ) k≤ e ( − β +127Λ) q | ρ | N + 2 e ( − β +260Λ) q | ρ | N ≤ e ( − β +260Λ) q | ρ | N . (cid:3) With the above preparation, we are in the place to prove Lemma 5.6. It is easy to see that all theestimates from (5.45) to (5.49) preserve errors between the traces. Now combine (5.45) with (5.47),we have (cid:12)(cid:12)(cid:12) | Tr A ( N q ) | − | Tr e A N ( q ) | (cid:12)(cid:12)(cid:12) ≤ e ( − β +260Λ) q | ρ | N ≤ | ρ | N , (5.52)provided e ( β − q > 4. Therefore, by (5.39), | Tr A ( N q ) | ≥ | Tr e A N ( q ) | − | ρ | N ≥ | ρ | N . (5.53)(5.49) implies that for any vector X ∈ C , k A − ( N q ) X k ≤ k A ( − N q ) X k + 4 e ( − β +260Λ) q | ρ | N k X k ≤ k A ( − N q ) X k + 18 | ρ | N k X k , (5.54)provided e ( β − q > By (5.8) and (5.12), it is easy to check that | det A ( N q ) | = | r − ( N q, | . Therefore, (5.15) impliesthat | det A ( N q ) | < e − ( β − q < Hu = Eu with normalized initial value X = (cid:18) u u (cid:19) , k X k =1. By (3.3) and the Cayley-Hamilton theorem for GL(2 , C ) matrix A ( N q ), we have:(5.56) A ( N q ) X = (cid:18) u Nq +1 u Nq (cid:19) , A ( − N q ) X = (cid:18) u − Nq +1 u − Nq (cid:19) . and A ( N q ) X + (det A ) · A − ( N q ) X = − (Tr A ( N q )) X (5.57)Combine (5.53),(5.54),(5.55) with (5.57), we have k A ( N q ) X k + k A ( − N q ) X k ≥ | ρ | N . (5.58)Now by the choice of ρ and N , for q large, we have k A ( N q ) X k + k A ( − N q ) X k ≥ 18 (1 + e − q ) [ e q ] ≥ e q (5.59)which implies max (cid:8) | u Nq +1 | , | u Nq | , | u − Nq +1 | , | u − Nq | (cid:9) ≥ e q . (5.60)In conclusion, we can claim the existence of a subsequence of u n at energy E with followingexponential growth: Claim 5.10. Assume v n , w n have β - q almost periodicity as in (2.1) and w n has (Λ , β ) - q bound(2.2),(2.2) for q > q (Λ , δ, β ) . Suppose E ∈ S q and β > (260 + δ )Λ , then there are integersequences x q , x q , x q , x q ∈ Z independent of E , such that min i | x iq | → ∞ as q → ∞ and max i | u Ex iq | > e q , (5.61) where u En solves the half-line problem Hu = Eu with normalized boundary condition | u | + | u | = 1 . Now Lemma 5.6 follows directly from Claim (5.10) and the following lemma: Theorem 5.11 (Extended Schnol’s Theorem, Lemma 2.4, [32]) . Fix any y > / . For any sequence | x k | → ∞ (where the sequence is independent of E ), for spectrally a.e. E , there is a generalizedeigenvector u E of Hu = Eu , such that | u Ex k | < C (1 + | k | ) y . The case where the trace is close to 2. In this part, we consider those energy E wherethe trace of A ( q ; E ) is close to 2. Let S q = { E : (cid:12)(cid:12) | Tr A ( q ; E ) | − (cid:12)(cid:12) < e − q } (5.62)Again we assume that q is large and v n , w n satisfy β - q almost periodicity (2.1) and Λ- q bound in(2.2) with positive finite parameters β, Λ , δ . We can prove that Lemma 5.12. If β > (130 + δ )Λ , then (5.63) µ ( S q ) < e − Λ q , where µ is the spectral measure of H . PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 27 Proof of Theorem 5.2: Assume now β > δ )Λ. Let q n be given as in Theorem 5.2. Lemma5.6 implies that for spectrally a.e. E , there is K ( E ) such that,(5.64) | Tr A ( q k ; E ) | < e − q k , ∀ k ≥ K ( E )Combine Lemma 5.12 with the Borel-Cantelli lemma, we have µ (cid:16) lim sup n S q n (cid:17) = 0, i.e., forspectrally a.e. E , there is K ( E ) such that(5.65) (cid:12)(cid:12)(cid:12) | Tr A ( q k ; E ) | − (cid:12)(cid:12)(cid:12) > e − q k , ∀ k ≥ K ( E ) . Clearly, (5.64) and (5.65) complete the proof of Theorem 5.2 by taking K = max { K , K } . (cid:3) In the rest of the section, we focus on proving (5.63). Similar to the hyperbolic case, Tr e A ( q ; E )and Tr A ( q ; E ) are close up to exponential error by Lemma 5.4. More precisely, let e S q := n E : (cid:12)(cid:12) | Tr e A ( q ; E ) | − (cid:12)(cid:12) < e − q o (5.66)Clearly, Lemma 5.4 implies that for β > S q ⊂ e S q .The following elementary linear algebra facts were proved in [32] Lemma 5.13 (Lemma 2.9, Lemma 2.10 [32]) . Suppose A ∈ SL(2 , R ) has eigenvalues ρ ± , ρ > .For any k ∈ N , if Tr A = 2 , then (5.67) A k = ρ k − ρ − k ρ − ρ − · (cid:16) A − Tr A · I (cid:17) + ρ k + ρ − k · I Otherwise, A k = k ( A − I ) + I .Assume further that (cid:12)(cid:12) | Tr A |− (cid:12)(cid:12) < τ < , then there are universal constants < C < ∞ , c > / such that for ≤ k ≤ τ − , we have (5.68) c < ρ k + ρ − k < C , c k < ρ k − ρ − k ρ − ρ − < C k. Now fix E ∈ e S q , the above lemma actually shows that the k -th power of e A ( q ; E ) grows almostlinearly with respect to k as : e A k ( q ) ∼ k (cid:0) e A ( q ) − 12 Tr e A ( q ) (cid:1) + I, ≤ k ≤ N. (5.69)This simple observation will be an important part of our quantitative estimates in the near paraboliccase. The arguments to derive (5.63) from (5.69) follow the outline of the near parabolic case in [32]with slight modification concerning all the estimates of the conjugacy in section 5.1. We sketch theproof below for reader’s convenience. Proof of Lemma 5.12 : First, Lemma 5.13 provides the following norm estimates: there is absoluteconstant C > ≤ j < N = [ e q ] < e q < e δβq , k e A j ( q ; E ) k < C j · k e A ( q ) k . (5.70)By (5.18) and the choice of N , we have k e A j ( q ; E ) k < C j e q < j e q < e q . (5.71)In the same way as the proof of (5.45) and (5.46), for any 1 ≤ k ≤ N , combine (5.71) with (5.27),we can apply Lemma 5.8 to obtain k e A ( kq ) − e A k ( q ) k ≤ e ( − β +130Λ) q < , (5.72) provided β > (130 + δ )Λ and q large.In view of (5.67), (5.69) and (5.72), it is clear that e A ( kq ) has the same linear expansion as in(5.69). Combine (5.67), (5.72) with the conjugate relation: A ( kq ) X = r − ( kq, T ( kq ) e A ( kq ) T − (0) X, X ∈ C , (5.73)we can prove that: Claim 5.14. For any ε > , E and ϕ ∈ [0 , π ) , let ℓ = ℓ ( ϕ, ε, E ) , u ϕ , v ϕ be given as in (3.21).Suppose E ∈ S q , ε < e − q and β > (130 + δ )Λ , then k u ϕ k ℓ > e Λ q . (5.74)The proof of Claim 5.14 follows the outline of the proof of Claim 5 in [32]. The key is to use thelinear expression (5.69) to control both the upper and lower bound of bound of k A ( n ) X k . The maindifference is we need to consider the conjugacy (5.73) and switch between the orbits of A ( n ) X and e A ( n ) ˜ X . We omit the details here. For sake of completeness, we include the proof in Appendix A.2.We proceed to prove Lemma 5.12 by Claim 5.14. Tr A ( q ; E ) is a polynomial in E with degree q . S q can be written as a union of at most q band: S q = S qj =1 I j . Note Tr e A ( q ; E ) is also a polynomialin E with degree q with real coefficients, by Proposition A.3, we have | e S q | ≤ C √ e − q , where C only depends on k w k ∞ , k v k ∞ . Then this gives us a uniform control on the width of each band I j :(5.75) S q = q [ j =1 I j , ε jq := | I j | ≤ | S q | ≤ | e S q | ≤ e − q . Now pick E j ∈ I j T σ ( H ) = ∅ to be the center in the sense that I j ⊂ ( E j − ε jq , E j + ε jq ). For any ϕ , let u ϕ ( E j ) be the right half line solution associated with the energy E j . By Claim 5.14, we have k u ϕ ( E j ) k ℓ q ( j ) ≥ e Λ q , j = 1 , · · · , q (5.76)where ℓ q ( j ) = ℓ ( ϕ, E j , ε jq ) is given as in (3.21).A direct consequence of (5.76) and the subordinacy theory Lemma 3.2 is ε jq · | m ϕ ( E j + iε jq ) | < √ · e − Λ q , j = 1 , · · · , q (5.77)Then by (2.3) and (3.25), we have µ ( I j ) ≤ sup ϕ ε jq | m ϕ ( E j + iε jq ) | < (5 + √ e − Λ q , j = 1 , · · · , q. (5.78)Clearly, (5.78) completes the proof of Lemma 5.12 provided q (5 + √ e − Λ q ≤ e − Λ q . (cid:3) Spectral Singularity for analytic quasiperiodic Jacobi operator In this section, we focus on analytic quasiperiodic potential given by v n = v ( θ + nα ) , w n = c ( θ + nα ) , n ∈ Z , θ ∈ T where v ∈ C ω ( T , R ) and c ∈ C ω ( T , C ) are analytic functions on T takenvalues in R and C respectively. Both v ( θ ) and c ( θ ) have bounded analytic extensions to the strip { z : | Im z | < ρ } .Follow the notations in section 3.1. We list the corresponding quasiperiodic versions here againfor reader’s convenience. The analytic quasiperiodic Jacobi operator on ℓ ( Z ) is given by:(6.1) ( H v,c u ) n = c ( θ + nα ) u n +1 + ¯ c ( θ + ( n − α ) u n − + v ( θ + nα ) u n , n ∈ Z . PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 29 The transfer matrix is given by: A ( θ, E, α ) = 1 c ( θ ) (cid:18) E − v ( θ ) − ¯ c ( θ − α ) c ( θ ) 0 (cid:19) and A ( n ; θ, E, α ) = n Y j =1 A (cid:0) θ + ( j − α, E, α (cid:1) , n > . The spectral singularity in Theorem 2.4 is reduced to the following lemma about the norm of thetransfer matrices, which was proved in [32]: Lemma 6.1 ([32], Lemma 3.1) . Fix α ∈ R \ Q with β = β ( α ) < ∞ and θ ∈ T . Suppose there is aconstant c > such that for any E , there is ℓ = ℓ ( E, β, ρ, θ ) such that for any ℓ > ℓ , the followingtwo estimates hold: (6.2) ℓ X k =1 k A ( k ; θ, E, α ) k ≥ ℓ cβ , and (6.3) ℓ X k =1 k A ( k ; θ − α, E, − α ) k ≥ ℓ cβ , then we have the following upper bound for the spectral dimension defined in (2.3) of the spectralmeasure µ = µ α,θ : dim spe ( µ ) ≤ γ := 11 + c/β < . (6.4)This is a direct consequence of the subordinate theory (3.23) and Last-Simon upper bound on thegeneralized eigenfunction (3.24). Actually, in view of Lemma 3.5, it is enough to find a ϕ such thatboth m ϕ and e m π/ − ϕ are γ -spectral singular, where m ϕ and e m π/ − ϕ are half line m-function definedin section 3.2. The estimate on the half line m -function relies on the subordinacy theory Lemma3.2. The quantitative estimates need both an upper bound and a lower bound on the ℓ -norm of u ϕ , v ϕ . Lemma 3.3 provides two eigen functions u ϕ and u ϕ, − , both obeying the sub-linear growth asin (3.24). (6.2) and (6.3) provide the lower bound as required in the subordinacy theory for m ϕ and e m π/ − ϕ respectively, which eventually lead to the spectral singularity. In the rest of this section, wewill focus on the proof of (6.2) and (6.3). We refer readers to [32], section 3 for more details aboutthis lemma and spectral singularity.For a GL(2 , C ) matrix A = (cid:18) a bc d (cid:19) , we denote by k · k HS the Hilbert-Smith norm of A : k A k HS = p | a | + | b | + | c | + | d | . (6.5)In the rest of this section, we write k · k = k · k HS for simplicity whenever it is clear.The key to prove (6.2) and (6.3) is the following lemma: Lemma 6.2. Assume that L ( E ) ≥ a > . There are c = c ( a, S, ρ ) > , n = n ( a, ρ ) > and apositive integer d = d ( S, ρ, k v k ρ , k c k ρ ) ∈ N + such that for E ∈ S and n > n , there exists an interval ∆ n ⊂ T satisfying the following properties: (6.6) Leb(∆ n ) ≥ c dn and for any θ ∈ ∆ n , (6.7) k A ( n ; θ, E, α ) k HS > e nL ( E ) / . Lemma 6.2 will be the key ingredient to the proof of spectral singularity, we will return to itsproof in the end of this section. We will derive (6.2) and (6.3) from Lemma 6.2 and finish the proofof Theorem 2.4 first.Let q n be given as in the continued fraction approximants to α , see (2.8). The following lemmaabout the ergodicity of an irrational rotation can be found e.g. in [25]. Lemma 6.3 (Lemma 9, [25]) . Let ∆ ⊂ [0 , be an arbitrary segment. If | ∆ | > q n . Then, for any θ ; there exists a j in { , , · · · , q n + q n − − } such that θ + jα ∈ ∆ . Combine Lemma 6.2 with Lemma 6.3, we immediately have the following localization densityresult: Lemma 6.4. Fix E ∈ S, θ ∈ Θ and α ∈ R \ Q . There is n = n ( E, ρ, α, θ ) such that for any q n ≥ n and any m ∈ N , there is j m = j m ( θ ) ∈ (cid:2) m q n , (2 m + 2) q n (cid:1) such that (6.8) k A ( j N ; θ, E, α ) k > e c q n L ( E ) where c = c ( a, ρ ) explicitly depends on c and d given in Lemma 6.2. Proof. We fix E , α and write A ( n ; θ ) = A ( n ; θ, E, α ) for simplicity. Let n be given as in Lemma6.2. Given q n , let(6.9) k n = [ c q n d ] − ≥ c d q n ≥ n , provided q n large, where c and d are given as in Lemma 6.2. By Lemma 6.2, there is an interval∆ k n ⊂ T such that the following hold:(6.10) Leb(∆ k n ) ≥ c dk n > q n and(6.11) k A ( k n ; θ ) k > e k n L ( E ) / > e c d q n L ( E ) , ∀ θ ∈ ∆ k n . Fix θ and m ∈ N , apply Lemma 6.3 to ∆ k n and θ + 2 m q n , we have that there exists a j in { , , · · · , q n + q n − − } such that ( θ + 2 m q n α ) + jα ∈ ∆ k n . By (6.11), we have k A ( k n ; θ + 2 m q n α + jα ) k > e c q n L ( E ) , (6.12)where c = c d .It is easy to check that A (2 m q n + j + k n ; θ ) = A ( k n ; θ + 2 m q n α + jα ) A (2 m q n + j ; θ ) . (6.13)By (6.12), we have that either k A − (2 m q n + j ; θ ) k ≥ e c q n L ( E ) (6.14) or k A (2 m q n + j + k n ; θ ) k ≥ e c q n L ( E ) . (6.15) PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 31 Direct computation shows that k A − (2 m q n + j ; θ ) k = 1 | det A (2 m q n + j ; θ ) | k A (2 m q n + j ; θ ) k (6.16) = | c ( θ + (2 m q n + j ) α ) || c ( θ ) | k A (2 m q n + j ; θ ) k (6.17) ≤ k c k ∞ | c ( θ ) | k A (2 m q n + j ; θ ) k (6.18)Suppose (6.14) holds, then k A (2 m q n + j ; θ ) k ≥ | c ( θ ) |k c k ∞ e c q n L ( E ) ≥ e c q n L ( E ) (6.19)provided e c q n L ( E ) ≥ k c k ∞ | c ( θ ) | . (6.20)Let j m be 2 m q n + j or 2 m q n + j + k n , for which j N satisfies (6.8). Clearly, by the choice of j, k n , j m ( θ ) ∈ (cid:2) m q n , (2 m + 2) q n (cid:1) for all m ∈ N and q n ≥ n := max (cid:8) dn c , ln k c k ∞ | c ( θ ) | c L ( E ) (cid:9) . (6.21)Note that if m = j = 0 in (6.12), we pick j = k n ≥ 1. So j ∈ (cid:2) , q n (cid:1) . (cid:3) With the above localization density lemma, we can complete the proof of Theorem 2.4 by checking(6.2) and (6.3) in Lemma 6.1 for a.e. θ ∈ T . Proof of Theorem 2.4. For any ℓ ∈ N , there is q n such that, l ∈ [2 q n , q n +1 ) . Let ℓ = 2 N q n + r, where 0 ≤ r < q n ,1 ≤ N < q n +1 q n . Let n be given as in (6.21). It is easy to check that q n ≥ n provided ℓ ≥ e n β ( α ) . (6.22)Now apply Lemma 6.4 to q n and 0 ≤ m ≤ N − 1. There are j m ∈ (cid:2) m q n , (2 m + 2) q n (cid:1) ⊂ [0 , ℓ ]such that k A ( j m ; θ, E, α ) k > e c q n L ( E ) . Therefore, ℓ X k =1 k A ( k ; θ, E, α ) k ≥ N − X m =0 k A ( j m ; θ, E, α ) k ≥ N e c q n L ( E ) . (6.23)Clearly, ℓ = 2 N q n + r < N q n . By (6.23), we have ℓ X k =1 k A k ( θ ) k ≥ ℓ q n e c q n L ( E ) ≥ ℓ e c q n L ( E ) ≥ ℓ e c a q n provided e c q n L ( E ) ≥ q n . Then for sufficiently large ℓ such that ln q n +1 q n < β , we have ℓ X k =1 k A k ( θ ) k ≥ ℓ q c a β n +1 ≥ ℓ · (cid:0) ℓ (cid:1) c a β ≥ ℓ · ℓ c a β =: ℓ cβ , (6.24)provided ℓ ≥ 4, where c = c a . This proved (6.2). For the same θ and E , repeat the above procedure for A ( n ; θ − α, E, − α ). We have a sequence ofpositive integers e j m = e j m ( θ − α ) ∈ (cid:2) mq n , m + 1) q n ) for any N ∈ N and q n ≥ n ( E, ρ, − α, θ − α )such that(6.25) k A ( e j m ; θ − α, E − α, E ) k > e c q n L ( E ) . Note that c = c ( a, ρ ) does not depend on θ − α and is the same as in (6.8) and (6.24). The samereasoning proves (6.3).Then by Lemma 6.1, we have for all θ ∈ Θ and β ( α ) < ∞ , dim spe ( µ α,θ ) < c/β < 1, whichcompletes the proof of Theorem 2.4. (cid:3) In the rest of the section, we focus on the proof of Lemma 6.2. In [32], the authors provedthe analytic SL (2 , R ) version of this lemma. One advantage for Shr¨odinger case is the H-S norm k A ( n ; θ ) k HS is a real analytic function which can be approximated by trigonometric functions insome uniform sense. For GL(2 , C ) case, the HS norm of the transfer matrices are meromorphicfunctions. We need finer decomposition to deal with the poles.Fix E, α , for n ∈ N + , θ ∈ T , let(6.26) F n ( θ ) = k A ( n ; θ, E, α ) k HS be defined as in (6.5). We have the following decomposition of F n ( θ ): Lemma 6.5. For any E and n ∈ N + , there are positive functions f n ( θ ) and g n ( θ ) such that F n ( θ ) = f n ( θ ) g n ( θ ) , (6.27) inf n n Z ln g n ( θ ) d θ = 0 , inf n n Z ln f n ( θ ) d θ = 2 L ( E ) . (6.28) For any ε > , there exists n = n ( ε ) ∈ N such that for any n > n and any θ ∈ T , < g n ( θ ) < e εn . (6.29) Furthermore, for E in a compact set S , there are n = n ( ρ ) > and d = d ( S, ρ, k v k ρ , k c k ρ ) > suchthat for any n > n , there are two functions P n ( θ ) , R n ( θ ) satisfying the following decomposition: f n ( θ ) = P n ( θ ) + R n ( θ ) , (6.30) | R n ( θ ) | < , (6.31) P n ( θ ) = X | k |≤ d · n b f n ( k ) e π i kθ , (6.32) where b f n ( k ) is the k -th Fourier coefficient of f n ( θ ) . Proof. Follow the notations in (3.4), let A ( θ, E ) = 1 c ( θ ) D ( θ, E ) , D ( θ, E ) = (cid:18) E − v ( θ ) − ¯ c ( θ − α ) c ( θ ) 0 (cid:19) and A ( n ; θ, E ) = 1 c ( n ; θ ) D ( n ; θ, E ) , where(6.33) c ( n ; θ ) = n Y j =1 c (cid:0) θ + ( j − α (cid:1) , D ( n ; θ, E ) = n Y j =1 D (cid:0) θ + ( j − α, E (cid:1) = (cid:18) D ( θ ) D ( θ ) D ( θ ) D ( θ ) (cid:19) (6.34) PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 33 Without loss of generality, we assume R T ln | c ( θ ) | d θ = 0. Otherwise, the argument simply differsby a constant factor. See Remark 6.1 after the proof.Let g n ( θ ) := (cid:12)(cid:12) c ( n ; θ ) (cid:12)(cid:12) and f n ( θ ) := k D ( n ; θ, E ) k HS . Clearly, F n ( θ ) = k A ( n ; θ, E, α ) k HS = f n ( θ ) g n ( θ )(6.35) f n ( θ ) = k D ( n ; θ, E ) k HS = | D ( θ ) | + | D ( θ ) | + | D ( θ ) | + | D ( θ ) | . (6.36)Birkhoff Ergodic Theory implies that for any irrational α ,lim n Z T n ln g n ( θ ) d θ = inf n n Z ln g n ( θ ) d θ = lim n Z T n n X j =1 ln | c (cid:0) θ + ( j − α (cid:1) | d θ = Z T ln | c ( θ ) | d θ = 0(6.37)In view of (6.35) and the definition of Lyapunov exponent (3.12), we haveinf n Z n ln f n ( θ ) d θ = inf n Z n ln (cid:16) g n ( θ ) k A ( n ; θ ) k HS (cid:17) d θ = inf n Z n ln g n ( θ ) d θ + inf n Z n ln k A ( n ; θ ) k HS d θ = 2 L ( E ) . (6.38)Note c ( θ ) is continuous in θ , by (3.14),for any ε > 0, there is n = n ( ε ) such that for any n > n and any θ ∈ T , we have the following upper semicontinuity (uniform in θ ):1 n ln g n ( θ ) ≤ Z T ln | c ( θ ) | d θ + ε = ε. (6.39)This gives g n ( θ ) ≤ e εn and finishes the proof of (6.27)-(6.29).The further decomposition of f n ( θ ) into P n and R n follows the strategy in [32]. Note that v ( θ )and c ( θ ) are both analytic with bounded extension to the strip { z : | Imz | < ρ } . In view of (6.34),all D i ( θ ) , i = 1 , , , { z : | Imz | < ρ } . For compact S , there is C = C ( S, ρ, k v k ρ , k c k ρ ) such that(6.40) k D i k ρ := sup | Imz | <ρ (cid:12)(cid:12)(cid:12) D i ( z ) (cid:12)(cid:12)(cid:12) < sup | Imz | <ρ k D n ( z ) k HS < e C n , E ∈ S, i = 1 , , , . Consider the Fourier expansion of the periodic-1 functions D i ( θ ):(6.41) D i ( θ ) = X k ∈ Z b D i ( k ) e πikθ , i = 1 , , , D i ( θ ) has exponential decay as | b D i ( k ) | < k D i k ρ · e − πρ | k | < e C n · e − πρ | k | , ∀ k ∈ Z , i = 1 , , , . (6.42)Combine(6.43) | D i ( θ ) | = (cid:0) X k ∈ Z b D i ( k ) e πikθ (cid:1) (cid:0) X k ∈ Z c D i ( k ) e − πikθ (cid:1) with (6.42), it is easy to check that the Fourier coefficients of | D i ( θ ) | has exponential decay as: (cid:12)(cid:12) \ | D i | ( · )( k ) (cid:12)(cid:12) < e C n · e − πρ | k | , ∀ k ∈ Z , i = 1 , , , . (6.44) Let f n ( θ ) be given as in (6.36). Consider the Fourier expansion of f n ( θ ):(6.45) f n ( θ ) = X k ∈ Z b f n ( k ) e πikθ . By (6.36) and (6.44), clearly, b f n ( k ) has the same exponential decay in | k | : | b f n ( k ) | < e C n · e − πρ | k | , ∀ k ∈ Z . (6.46)Pick(6.47) d = (cid:20) C πρ (cid:21) + 2 . We split f n ( θ ) into two parts: f n ( θ ) = P n ( θ ) + R n ( θ ) , P n ( θ ) = X | k |≤ d · n b f n ( k ) e πikθ , R n ( θ ) = X | k | >d · n b f n ( k ) e πikθ . For any θ ∈ T , | R n ( θ ) | ≤ X | k | >d · n | b f n ( k ) | ≤ X | k | >d · n e C n · e − πρ | k | ≤ − e − πρ e C n e − πρdn ≤ − e − πρ e − ( πρ d − C ) n . By the choice of d in (6.47), we have πρd > C + πρ . Then for any θ ∈ T ,(6.48) | R n ( θ ) | ≤ − e − πρ e − πρ n < , provided n > n ( ρ ) := ( πρ ) − ln( − e − πρ ). This finishes the proof of (6.30)-(6.32). (cid:3) Remark . Suppose b = R T ln | c ( θ ) | d θ = 0. In (6.33), we set A ( θ, E ) = 1 e c ( θ ) e D ( θ, E ) , where e c ( θ ) = e − b c ( θ ) , e D ( θ, E ) = e − b D ( θ, E ) . (6.49)Clearly, Z T ln | e c ( θ ) | d θ = 0 , lim n Z T n ln k e D ( n ; θ, E ) k d θ = L ( E ) . (6.50)Let g n ( θ ) := (cid:12)(cid:12)e c ( n ; θ ) (cid:12)(cid:12) and f n ( θ ) := k e D ( n ; θ, E ) k HS . The rest of the decomposition are exactly thesame.Combine Lemma 6.5 with the positive assumption on Lyapunov exponent, we can now finish The proof of Lemma 6.2: Assume that the Lyapunov exponent L ( E ) ≥ a > E ∈ S . Pick ε = a/ 8. Let n = n ( ε ) and n = n ( ρ ) be given as in Lemma 6.5. Then for all n > max { n , n } ,we have g n ( θ ) , f n ( θ ) , P n ( θ ) and R n ( θ ) as in Lemma 6.5, satisfying (6.27)-(6.32). DenoteΘ n = { θ : F n ( θ ) > e nL ( E ) / } , Θ n = { θ : P n ( θ ) > e nL ( E ) / } , Θ n = { θ : f n ( θ ) > e nL ( E ) / } . PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 35 Let n := 4 a − . Then for all n > n , we have e nL ( E ) > e na > e > 50. By using the fact x / − x / > x / − x / > x > 50, it is easy to check that for n > n , e nL ( E ) / − e nL ( E ) / > e nL ( E ) / − e nL ( E ) / > . (6.51)Assume that f n ( θ ) > e nL ( E ) / . By (6.30) and (6.51), we have for n > n , P n ( θ ) > f n ( θ ) − | R n ( θ ) | > e nL ( E ) / − > e nL ( E ) / . Then f n ( θ ) > P n ( θ ) − | R n ( θ ) | > e nL ( E ) / − > e nL ( E ) / . In view of (6.27) and (6.29), we have then for n > max { n , n } , F n ( θ ) = f n ( θ ) g n ( θ ) > e nL ( E ) / e nε > e nL ( E ) / e nL ( E ) / = e nL ( E ) / . Therefore, we have for n > n := max { n , n , n } ,(6.52) Θ n ⊆ Θ n ⊆ Θ n . Meanwhile, by (6.28),2 nL ( E ) ≤ Z T ln f n ( θ )d θ ≤ Leb(Θ n ) ln k f n k ρ + (cid:0) − Leb(Θ n ) (cid:1) ln e nL ( E ) / ≤ Leb(Θ n ) · C n + (cid:0) − Leb(Θ n ) (cid:1) · nL ( E ) / . This implies Leb(Θ n ) ≥ L ( E )2 C − L ( E ) . Note that L ( E ) ≥ a > , E ∈ S , we have(6.53) Leb(Θ n ) ≥ a C − a =: c ( a, S, ρ ) > . In view of (6.52), we have for n > n ,(6.54) Leb(Θ n ) ≥ c ( a, S, ρ ) > . By (6.32), P n ( θ ) is a trigonometric polynomial of degree (at most) 2 dn , where d is given by (6.47) inLemma 6.5. The set Θ n consists of no more than 4 dn intervals. Therefore, there exists a segment,∆ n ⊂ Θ n ⊂ Θ n , with Leb(∆ n ) > c dn . For any n > n and θ ∈ ∆ n ⊂ Θ n , k A n ( θ ) k HS = F n ( θ ) > e nL ( E ) / and Leb(∆ n ) > c dn , as claimed. (cid:3) The Extended Harper’s model: proof of Corollary 2.5 Recall the extended Harper’s model (EHM) defined in (2.13) as:(7.1) ( H λ,α,θ u ) n = c λ ( θ + nα ) u n +1 + ¯ c λ (cid:0) θ + ( n − α (cid:1) u n − + 2 cos 2 π ( θ + nα ) u n , where c λ ( θ ) = λ e − πi ( θ + α ) + λ + λ e πi ( θ + α ) . (7.2)By some earlier work [28], we consider the following partitioning of the parameter space into thefollowing three regions: λ λ + λ λ + λ = λ 11 Region I Region IIRegion III L II L I L III Region I: ≤ λ + λ ≤ , < λ ≤ Region II: max { λ + λ , } ≤ λ , λ + λ > Region III: max { , λ } ≤ λ + λ , λ > L ( E, λ ) be the Lyapunov exponent of the extended Harper’s model, defined as in (3.12). Themain achievement of [28] is to prove the following explicit formula of L ( E, λ ), valid for all λ and allirrational α : Theorem 7.1 ([28]) . Fix an irrational frequency α . Then L ( E, λ ) restricted to the spectrum is zerowithin both region II and III. In region I it is given by the formula on the spectrum, (7.3) L ( E, λ ) = ln (cid:18) √ − λ λ λ (cid:19) , if λ ≥ λ , λ ≤ λ + λ , ln (cid:18) √ − λ λ λ (cid:19) , if λ ≥ λ , λ ≤ λ + λ , ln √ − λ λ λ + p λ − λ λ ! , if λ ≥ λ + λ . Denote by Region I ◦ ,Region II ◦ , Region III ◦ the interior of Region I,II,III respectively. A com-plete understanding of the spectral properties of the extended Harper’s model for a.e. θ has beenestablished in [23, 22, 3, 20]. We collect the spectral decomposition results in these papers as thefollow theorem for reader’s convenience. Follow the notations in Corollary 2.5, denote the threeparameter regions of λ = ( λ , λ , λ ) ∈ R by: R = (cid:8) λ ∈ R : 0 < λ + λ < , < λ < (cid:9) . R = (cid:8) λ ∈ R : λ > max { λ + λ , } , λ + λ ≥ λ + λ > max { λ , } , λ = λ , λ > (cid:9) . R = { λ ∈ R : 0 ≤ λ + λ ≤ , λ = 1 or λ + λ ≥ max { λ , } , λ = λ , λ > } . Theorem 7.2 ([23, 22, 3, 20]) . The following Lebesgue decomposition of the spectrum of H λ,α,θ holds for a.e. θ . • For λ ∈ R , if β ( α ) < L ( E, λ ) , then H λ,α,θ has pure point spectrum. If β ( α ) > L ( E, λ ) ,then H λ,α,θ has purely singular continuous spectrum. PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 37 • For λ ∈ R and all irrational α , H λ,α,θ has purely absolutely continuous spectrum. • For λ ∈ R and all irrational α , H λ,α,θ has purely singular continuous spectrum. Now we are in the place to analyze the spectral dimension of EHM in each region.Clearly, Region I ◦ = R . In view of (7.3), it is easy to check that L ( E, λ ) > R for all α and E . Therefore, by Theorem 2.3, we have part (1) of Corollary 2.5.Next, consider Region R . Theorem 7.2 shows that H λ,α,θ has purely a.c. spectrum in region R for all α and a.e. θ . In view of Definition 2.3, absolutely continuous measure has full spectraldimension . This gives part (2) of Corollary 2.5.Part (3) (region R ) is the only place requires extra work. By Theorem 7.1 and Theorem 7.2, inregion R , L ( E, λ ) = 0 on the spectrum and H λ,α,θ does not have a.c. spectrum. Lack of positivityof Lyapunov exponent, we do not have the spectral singularity and the upper bound provided byTheprem 2.4. While the lower bound from Theorem 2.1 still holds. Moreover, in view of Lemma 3.1and Corollary 4.3, we can obtain arbitrarily small exponential growth of the transfer matrix. Thisallows us to obatian the increased range of β ( α ) in the critical region in part (3).Recall the notations of the tranfer matrix in (3.6) and (3.7) for EHM: let A λ ( θ, E, α ) = 1 c λ ( θ ) D λ ( θ, E, α ) , D λ ( θ, E, α ) = (cid:18) E − v ( θ ) − ¯ c λ ( θ − α ) c λ ( θ ) 0 (cid:19) . For n > , m ∈ Z , A λ ( n, m ; θ ) = m + n − Y j = m A λ (cid:0) θ + jα (cid:1) , (7.4) D λ ( n, m ; θ ) = m + n − Y j = m D λ (cid:0) θ + jα (cid:1) , c λ ( n, m ; θ ) = m + n − Y j = m c λ (cid:0) θ + jα (cid:1) . (7.5)It is easy to check that L ( E, λ ) = L ( D λ ) − Z T ln | c λ ( θ ) | d θ. (7.6)Note that b λ := Z T ln | c λ ( θ ) | d θ (7.7)is not necessarily zero in region R . Suppose not, consider the rescaling trick in Remark 6.1. Set e c λ ( θ ) = e − b λ c λ ( θ ) , e D λ ( θ, E ) = e − b λ D λ ( θ, E ) . (7.8)Clearly, in Region X, Z T ln | e c ( θ ) | = 0 , L ( e D λ ) = L ( D λ ) − b λ = L ( E, λ ) = 0 . (7.9)Let e D λ ( n, m ; θ ) , e c λ ( n, m ; θ ) be defined the same way as in (7.5). For irrational α , let β ( α ) and q n be defined as in (2.8). Now assume β ( α ) > 0, let e β = min { β ( α ) / , } . It was proved in [28] that L ( E, α ) is continuous in E for irrational α . In view of Lemma 3.1, the lim sup is uniform in both θ Actually, it is well known that a.c. measure has full dimension for most commonly used fractal dimensions, e.g.Hausdorff/packing dimension etc. See more background knowledge about fractal dimension in e.g. [14] We still denote the subsequence reaching the lim sup by q n . and E . Therefore, for any δ > 0, there is n = n ( δ, e β ) such that for any n > n , m ∈ Z , θ ∈ T and E ∈ σ ( H λ,α,θ ), k e D λ ( n, m ; θ ) k ≤ e δ e β n , (7.10) | e c λ ( n, m ; θ ) | ≤ e δ e β n . (7.11)Note that in the proof Theorem 5.2, we only need to consider the above upper bound for E restrictedin the spectrum. By Corollary 4.3, for a.e. θ and q n large,min | m |≤ e δβ qn | e c λ ( q n , m ; θ ) | > e − δ e β q n . (7.12)Combing (7.10), (7.11) and (7.12), exact the same computation in section 3.4 shows that for a.e. θ ,0 < δ < √ and q n large,min | m |≤ e δβ qn | e c λ ( r, m ; θ ) | ≥ e − δ e β q n , ≤ r ≤ q n , (7.13) max | m |≤ e δβ qn (cid:12)(cid:12)(cid:12) e c λ ( θ + ( m ± q n ) α ) e c λ ( θ + mα ) − (cid:12)(cid:12)(cid:12) < e − ( β − δ e β ) q n , (7.14) sup E ∈ σ ( H λ,α,θ ) k A λ ( r, m ; θ ) k < e δ e βq n , ≤ r ≤ q n , | m | ≤ e δβ q n . (7.15)Therefore, we can replace all the Λ in the proof Theorem 5.2 by 10 δ e β . Then for any β ( α ) > < γ < 1, (5.4) holds true provided δ < − γ ) . (7.16)Thereofore, by Lemma 5.1 and Theorem 5.2, for any β ( α ) > γ < θ , µ λ,α,θ is γ -spectralcontinuous. By (2.5), dim spe ( µ λ,α,θ )=1, which completes the proof of part (3) of Corollary 2.5. (cid:3) Appendix A. Appendix A.1. Proof of (5.4) in Theorem 5.2. We have showed in the first part of Theorem 5.2 that if β > δ )Λ, then for µ a.e. E , there exists K ( E ) ∈ N , for k ≥ K ( E ), we have | T rA ( q k ; E ) | < − e − q k (A.1)Now by (5.20), we have | T r e A ( q k ; E ) | < − e − q k + 12 e ( − β +4Λ) q k < − e − q k , (A.2)provided e ( β − q k > 12. Fix E and q = q k and write e A ( q k ; E ) = e A ( q ). Now apply Lemma (5.13)to these e A ( q ) satisfying A.2. Note e A ( q ) ∈ SL(2 , R ), and | Tr e A ( q ) | < 2, the eigenvalue ρ of e A ( q ) ispurely imaginary with modulus 1, i.e., ρ = e iψ , for some ψ ∈ ( − π, π ). By (5.67), we have for any j ,(A.3) e A j ( q ) = sin jψ sin ψ · (cid:16) e A ( q ) − Tr e A ( q )2 · I (cid:17) + cos jψ · I, ψ ∈ ( − π, π )Then | ψ | = | Tr e A ( q ) | < − e − q implies | sin ψ | > q − (1 − e − q ) > e − q . By (A.3)and (5.18), k e A j ( q ) k ≤ e q k e A ( q ) k + 1 ≤ e q , (A.4)provided q > q (Λ). PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 39 Now for any 0 < γ < 1, let ξ = − γ < e δβq and N = [ e ξ Λ q ] . (A.5)Apply Lemma 5.8 to G = e A ( q ), G j = e A ( q, jq + 1), j = 0 , · · · , N , by (5.27) and (A.4), for all j ≤ N we have k e A ( jq ) − e A j ( q ) k < e (cid:0) − β +93Λ+2 ξ Λ (cid:1) q < e − Λ q < . (A.6)provided β > (94 + 2 ξ )Λ. Therefore, by (A.4), k e A ( jq ) k ≤ k e A j ( q ) k + 1 ≤ e q k e A ( q ) k + 2 ≤ e q By (5.15), | r − ( jq, | ≤ N e ( − β +2Λ) q ≤ e ( − β +2Λ+ ξ Λ) q < β > 3Λ + ξ Λ. Then by(5.12) and (5.18), for all 0 ≤ j ≤ N and 1 ≤ r ≤ q , k A ( jq ) k ≤ | r − ( jq, | · k T jq k · k e A ( jq ) k · k T − k ≤ e q (A.7) k A ( jq + r ) k ≤ k A ( r, jq + 1) k · k A ( jq ) k ≤ e q (A.8)Therefore, Nq X n =1 k A ( n ; E ) k ≤ N − X k =0 q X r =1 k A ( kq + r ; E ) k ≤ N q e q ≤ e ( ξ +93)Λ q (A.9) 1( N q ) − γ Nq X n =1 k A ( n ; E ) k ≤ e (cid:0) − (1 − γ ) ξ +94 (cid:1) Λ q = e − Λ q < < γ < µ a.e. E , we have a sequence q k → ∞ and ℓ k = [ e − γ ) − Λ q k ] q k such that ℓ k X n =1 k A ( n ; E ) k ≤ ℓ − γk (A.11)provided β > (3 ξ + ξ/δ )Λ = (285 + 95 δ ) Λ1 − γ > (94 + 2 ξ + ξ/δ )Λ . (A.12)It was proved in [32] that (A.11) implies (5.2) directly from the relation (3.3) and (3.21) and . Weomit the proof for this part here. See more details about this direct computation in the proof Lemma2.1 in [32]. (cid:3) A.2. Proof of Claim 5.14. For any 0 < ε < e − q , let ℓ = ℓ ( ϕ, ε, E ) , u ϕ , v ϕ be given as in (3.21).Write ℓ ( ε ) = [ ℓ ] + ℓ − [ ℓ ], and [ ℓ ] = K ( ε ) · q + r ( ε ), where 0 ≤ r = [ ℓ ]mod q < q and 0 ≤ ℓ − [ ℓ ] < X = (cid:18) cos ϕ − sin ϕ (cid:19) and e X = T − X . Clearly, k X k = k e X k = 1.We need to show first suppose K < N q = [ e q ], then for any ε < e − q :(A.13) K > e Λ q For any n ≤ [ ℓ ] + 1, write n = kq + r , where 0 ≤ k ≤ K, ≤ r ≤ q . By (5.15), (5.71) and (5.72),we have k A ( kq ) k ≤ | r − ( kq, | · ( k e A k ( q ) k + 1) ≤ C k e q + 1) < k e q Then by (5.18), k A ( kq + r ) X k ≤ k A ( r, kq + 1) k · k A ( kq ) k · k X k ≤ k e q Direct computation shows k u ϕ k ℓ ≤ [ ℓ ]+1 X n =1 k A ( n ) · X k ≤ q X r =1 k A ( r ) · X k + K X k =1 q X r =1 k A ( kq + r ) · X k ≤ q · e q + K X k =1 q X r =1 k e q ≤ q · e q + K q e q ≤ K e q Since ϕ is arbitrary, we have k v ϕ k ℓ ≤ K e q in the same way. By the defintion of ℓ in (3.21),we have(A.14) K e q ≥ k u ϕ k ℓ ( ε ) k v ϕ ( ε ) k ℓ = 12 ε ≥ e q Therefore, K > e Λ q as claim in (A.13).To bound k u ϕ k ℓ from below, we need to consider two cases of initial value ϕ . Case I: Assume ϕ satisfies(A.15) k (cid:0) e A ( q ) − Tr e A ( q )2 · I (cid:1) · e X k ≥ e − Λ q . By (5.67), for any e Λ q ≤ k ≤ K ≤ N q , we have k e A k ( q ) · e X k ≥ ρ k − ρ − k ρ − ρ − · k (cid:16) e A ( q ) − Tr e A ( q )2 · I (cid:17) e X k − ρ k + ρ − k · k e X k≥ k · e − Λ q − C ≥ , provided e Λ q > C + 3).By (5.72), we have then k e A ( kq ) · e X k ≥ k e A k ( q ) · e X k − k (cid:16) e A ( kq ) − e A k ( q ) (cid:17) · e X k ≥ . By (5.12) and (5.15), for e Λ q ≤ k ≤ K , we have(A.16) k A ( kq ) X k = | r − ( kq, | · k T kq e A ( kq ) T − X k = | r − ( kq, | · k e A ( kq ) e X k ≥ k u ϕ k ℓ ≥ [ ℓ ] − X n =1 k A n · X k ≥ X e 14 Λ q ≤ k ≤ K k A ( kq ) · X k ≥ 12 ( K − e Λ q ) > e Λ q Case II: Assume ϕ satisfies(A.17) k (cid:0) e A ( q ) − Tr e A ( q )2 I (cid:1) · e X k < e − Λ q , PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 41 By (5.67), for any 1 ≤ k ≤ e Λ q < N q we get k e A k ( q ) · e X k ≥ ρ k + ρ − k · k e X k − ρ k − ρ − k ρ − ρ − · k (cid:16) e A ( q ) − Tr e A ( q )2 I (cid:17) e X k≥ − C k · e − Λ q ≥ , provided e Λ q > C .By (5.72), we have k e A ( kq ) · e X k ≥ k e A k ( q ) · e X k − k ( e A ( kq ) − e A k ( q )) · e X k ≥ . By (5.12) and (5.15), for any 1 ≤ k ≤ e Λ q < N q , we have(A.18) k A ( kq ) X k = | r − ( kq, | · k T kq e A ( kq ) T − X k = | r − ( kq, | · k e A ( kq ) e X k ≥ k u ϕ k ℓ ≥ [ ℓ ] − X n =1 k A n · X k ≥ X ≤ k ≤ e 15 Λ q k A ( kq ) · X k ≥ e Λ q ≥ e Λ q . (cid:3) A.3. The refined estimate on the preimage of P n ( R ) . 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. Thefollowing proposition was proved in Theorem 6.1,[29]: Proposition A.1. Let p ∈ P n ; n ( R ) with y < · · · < y n − the local extrema of p . Let (A.19) ζ ( p ) := min ≤ j ≤ n − | p ( y j ) | and ≤ a < b . Then, | p − ( a, b ) | ≤ diam ( z ( p − a )) max n b − aζ ( p ) + a , (cid:0) b − aζ ( p ) + a (cid:1) o (A.20) where z ( p ) is the zero set of p and | · | denotes the Lebesgue measure. Acknowledgement The authors would like to thank Svetlana Jitomirskaya for reading the early manuscript andvaluable suggestions. The authors would also like to thank Ilya Kachkovskiy for useful comments.R. H. and F. Y. would like to thank the Institute for Advanced Study, Princeton, for its hospitalityduring the 2017-18 academic year. R. H. and F. Y. were supported in part by NSF grant DMS-1638352. Research of S. Z. was supported in part by NSF grant DMS-1600065. References 1. A. Avila and S. Jitomirskaya, The ten martini problem , Annals of Mathematics 170, 303-342 (2009).2. A. Avila and S. Jitomirskaya, H¨older continuity of absolutely continuous spectral measures for one-frequencySchr¨odinger operators. Commun. Math. Phys. 301, 563-581 (2011).3. A. Avila, S. Jitomirskaya and C. Marx, Spectral theory of Extended Harpers Model and a question by Erds andSzekeres , Inventiones mathematicae 210.1, 283-339 (2017).4. A. Avila, J. You and Q. Zhou, Sharp phase transitions for the almost Mathieu operator . Duke Math. J, 166.14,2697-2718 (2017). 5. J. Barbaroux, F. Germinet and S. Tcheremchantsev, dimensions and the phenomenon of intermittency in quan-tum dynamics. Duke Math. J. 110, 161-193 (2001).6. S. Becker, R. Han and S. Jitomirskaya, Cantor spectrum of graphene in magnetic fields, preprint arXiv:1803.00988.7. R. Carmona and J. Lacroix, Spectral theory of random Schr¨odinger operators. Springer Science & Business Media,(2012).8. D. Damanik, Lyapunov exponents and spectral analysis of ergodic Schr¨odinger operators: a survey of Kotanitheory and its applications. Proc. Sympos. Pure Math. 76, Part 2, Providence, RI: Amer. Math. Soc., 2007, pp.5395639. D. Damanik, Schr¨odinger operators with dynamically defined potentials. Ergodic Theory and Dynamical Systems37(6), 1681-1764 (2017).10. D. Damanik, A. Grodetski and W. Yessen. The Fibonacci Hamiltonian. Inventiones mathematicae 206.3 629-692(2016).11. D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals. iii. α -continuity. Commun. Math. Phys. 212, 191-204 (2000).12. D. Damanik and S. Tcheremchantsev, Upper bounds in quantum dynamics. J. Amer. Math. Soc. 20, 799-827(2007).13. R. del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum, IV. Hausdorffdimensions, rank one perturbations, and localization. J. Anal. Math. 69, 153-200 (1996).14. K. Falconer, Techniques in Fractal Geometry , John Wiley & Sons, Ltd., Chichester, (1997).15. A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems Annales de l’Institut HenriPoincare (B) Probability and Statistics. No longer published by Elsevier, 33(6): 797-815 (1997).16. D.J. Gilbert, On subordinacy and analysis of the spectrum of Schr¨odinger operators with two singular endpoints. Proc. Roy. Soc. Edinburgh A 112, 213-229 (1989).17. F. Germinet, A. Kiselev and S. Tcheremchantsev, Transfer matrices and transport for 1D Schr¨odinger operatorswith singular spectrum . Ann. Inst. Fourier 54, 787-830 (2004)18. D.J. Gilbert and D. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schr¨odingeroperators. J. Math. Anal. Appl. 128, 30-56 (1987).19. I. Guarneri and H. Schulz-Baldes, Lower bounds on wave packet propagation by packing dimensions of spectralmeasures. Math. Phys. Electron. J. 5(1), 16 (1999).20. R. Han. Absence of point spectrum for the self-dual extended Harpers model . Int. Math. Res. Not. rnw279 (2017).21. R. Han. Dry ten martini problem for the non-self-dual extended Harpers model . Transactions of the AmericanMathematical Society, 370.1, 197-217 (2018).22. R. Han and S. Jitomirskaya, Full measure reducibility and localization for quasi-periodic Jacobi operators: atopological criterion. Advances in Mathematics 319, 224-250 (2017).23. S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Localization for a Family of One-dimensional Quasi-periodicOperators of Magnetic Origin , Ann. Henri Poincar`e 6, 103-124 (2005).24. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I. Half-line operators. Acta Math.183, 171-189 (1999).25. S. Jitomirskaya and Y. Last. Power-law subordinacy and singular spectra. II. Line operators. Commun. Math.Phys. 211, 643-658 (2000).26. S. Jitomirskaya and W. Liu, Universal hierarchical structure of quasiperiodic eigenfunctions. Annals of Math, toappear.27. S. Jitomirskaya and W. Liu, Arithmetic Spectral Transitions for the Maryland Model. Comm. Pure Appl. Math.,70: 1025-1051 (2017).28. S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponentof Extended Harper’s Model , Commun. Math. Phys. 316, 237-267 (2012).29. S. Jitomirskaya and C.A. Marx, Analytic quasi-periodic Schr¨odinger operators and rational frequency approxi-mants , Geom. Funct. Anal. 22, 1407-1443 (2012).30. S. Jitomirskaya and R. Mavi, Dynamical bounds for quasiperiodic Schr¨odinger operators with rough potentials. Int. Math. Res. Not. 1, 96-120 (2017).31. S. Jitomirskaya and F. Yang, Singular continuous spectrum for singular potentials. Communications in Mathe-matical Physics 351.3 1127-1135 (2017).32. S. Jitomirskaya and S. Zhang, Quantitative continuity of singular continuous spectral measures and arithmeticcriteria for quasiperiodic Schr¨odinger operators arXiv:1510.07086 (2015).33. S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensionalSchr¨odinger operators , Stochastic Analysis (Katata/Kyoto, 1982), 225-247, North-Holland Math. Library 32,North-Holland, Amsterdam, 1984 PECTRAL DIMENSION FOR SINGULAR JACOBI OPERATORS 43 34. R. Killip, A. Kiselev and Y. Last, Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125,1165-1198 (2003).35. Y. Last, Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142, 406-445(1996).36. Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensionalSchrdinger operators. Inventiones mathematicae 135.2, 329-367 (1999).37. C. A. Marx,
τ − δ > τ − δ . The latter gives the restriction on δ suchthat δ < τ − .By Lemma 4.1, q n − Y j =0 ,j = j l,k | sin π ( θ − θ ℓ + kq n α + jα ) | ≥ e − ( q n −