Absolutely Continuous Spectrum for CMV Matrices With Small Quasi-Periodic Verblunsky Coefficients
aa r X i v : . [ m a t h . SP ] F e b ABSOLUTELY CONTINUOUS SPECTRUM FOR CMVMATRICES WITH SMALL QUASI-PERIODICVERBLUNSKY COEFFICIENTS
LONG LI, DAVID DAMANIK, AND QI ZHOU
Abstract.
We consider standard and extended CMV matrices withsmall quasi-periodic Verblunsky coefficients and show that on their es-sential spectrum, all spectral measures are purely absolutely continuous.This answers a question of Barry Simon from 2005. Introduction
This paper is concerned with the spectral analysis of (standard and ex-tended) CMV matrices with quasi-periodic Verblunsky coefficients. CMVmatrices are canonical matrix representations of unitary operators with acyclic vector, and they arise naturally in the context of orthogonal polyno-mials on the unit circle. We refer the reader to [40, 41] for background.Let us recall how CMV matrices arise in connection with orthogonal poly-nomials on the unit circle. Suppose µ is a non-trivial probability measureon the unit circle ∂ D = { z ∈ C : | z | = 1 } , that is, µ ( ∂ D ) = 1 and µ is notsupported on a finite set. By the non-triviality assumption, the functions 1, z , z , · · · are linearly independent in the Hilbert space H = L ( ∂ D , dµ ), andhence one can form, by the Gram-Schmidt procedure, the monic orthogonalpolynomials Φ n ( z ), whose Szeg˝o dual is defined by Φ ∗ n = z n Φ n (1 /z ). Thereare constants { α n } ∞ n =0 in D = { z ∈ C : | z | < } , called the Verblunskycoefficients , so that(1.1) Φ n +1 ( z ) = z Φ n ( z ) − α n Φ ∗ n ( z ) , which is the so-called Szeg˝o recurrence . Conversely, every sequence { α n } ∞ n =0 in D arises in this way.The orthogonal polynomials may or may not form a basis of H . However,if we apply the Gram-Schmidt procedure to 1 , z, z − , z , z − , . . . , we willobtain a basis – called the CMV basis . In this basis, multiplication by the
D.D. was supported in part by NSF grant DMS–1700131, an Alexander von HumboldtFoundation research award, and a Simons Fellowship.Q.Z. was supported by National Key R&D Program of China (2020YFA0713300),NSFC grant (12071232), The Science Fund for Distinguished Young Scholars of Tianjin(No. 19JCJQJC61300) and Nankai Zhide Foundation. independent variable z in H has the matrix representation C = α α ρ ρ ρ · · · ρ − α α − ρ α · · · α ρ − α α α ρ ρ ρ · · · ρ ρ − ρ α − α α − ρ α · · · α ρ − α α · · ·· · · · · · · · · · · · · · · · · · , where(1.2) ρ n = (1 − | α n | ) / for n ≥
0. A matrix of this form is called a
CMV matrix .It is sometimes helpful to also consider a two-sided extension of a matrixof this form. Namely, given a bi-infinite sequence { α n } n ∈ Z in D (and definingthe ρ n ’s as before), we may consider the extended CMV matrix E = · · · · · · · · · · · · · · · · · · · · ·· · · − α α − α ρ ρ ρ · · ·· · · − ρ α − − α α − ρ α · · ·· · · α ρ − α α α ρ ρ ρ · · ·· · · ρ ρ − ρ α − α α − ρ α · · ·· · · α ρ − α α · · ·· · · · · · · · · · · · · · · · · · · · · . Naturally, one is interested in both direct and inverse spectral results,depending on whether one starts with information about the Verblunskycoefficients or the measure. This paper is concerned with a direct spectralproblem. The Verblunsky coefficients will be small and quasi-periodic, andour goal is to show that the associated spectral measures are purely ab-solutely continuous (on the essential spectrum). We aim to establish thisproperty for both standard and extended CMV matrices. This addresses oneof the open problems described by Simon in [41]. Indeed, in the Remarksand Historical Notes to [41, Section 10.16], he writes that from his discus-sion of ergodic Verblunsky coefficients “conspicuously absent is the case ofalmost periodic Verblunsky coefficients” and “especially interesting is thequasiperiodic case.” He goes on to suggest that one should prove absolutecontinuity for small quasi-periodic coefficients and pure point spectrum forsome quasi-periodic examples. We note that quasi-periodic examples withpure point spectrum were exhibited by Wang-Damanik in [45], making useof earlier results of Zhang [50] on the positivity of the Lyapunov exponent inthe setting in question. Here we address the first part of Simon’s question,namely how to prove absolute continuity of the spectral measures for smallquasi-periodic Verblunsky coefficients.Let us now describe the setting and the main result in detail. We considersmall analytic quasi-periodic Verblunsky coefficients of the form(1.3) α n ( x ) = α ( x + ( n − ω ) , n ∈ Z , .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 3 where(1.4) α ( x ) = λe πih ( x ) ,h ∈ C ω ( T d , R ), λ ∈ (0 , ω ∈ R d with h m, ω i / ∈ Z for any m ∈ Z d \{ } .The sequence in (1.3)–(1.4) defines an extended CMV matrix. Since thismatrix formally depends on x ∈ T d , we will denote it by E x . While it ofcourse also depends on h and ω , we view them as fixed and suppress themfrom the notation. To define a standard CMV matrix, we only consider thevalues of α n ( x ) in (1.3) for n ∈ Z + and denote the resulting one-sided CMVmatrix by C x .Recall that ω ∈ R d is called Diophantine if there exist some κ, τ > j ∈ Z |h n, ω i − j | ≥ κ | n | τ for all n ∈ Z d \{ } . Let DC( κ, τ ) be the set of all Diophantine numbers withprescribed κ, τ .We can now state our main result: Theorem 1.1.
Suppose that r, κ, τ > , h ∈ C ωr ( T d , R ) , and ω ∈ DC( κ, τ ) .Then there exists λ = λ ( r, κ, τ ) > such that for λ ∈ (0 , λ ) , the followingholds for the Verblunsky coefficients given by (1.3) – (1.4) : • For every x ∈ T d , E x has purely absolutely continuous spectrum. • For every x ∈ T d , the restriction of the canonical spectral measureof C x to Σ = σ ess ( C x ) is purely absolutely continuous and C x has atmost one eigenvalue in each connected component of ∂ D \ Σ . Remark 1.2. (a) The set of all Diophantine ω ∈ R d has full Lebesguemeasure. It is an interesting question whether the conclusion of the theoremcan fail for some ω ’s, even if λ is small. In the Schr¨odinger setting, it isknown that the cases d = 1 and d > λ > κ and τ and the result then holds for all Diophantine ω withthis choice of λ ) in the case d = 1 [3], and it is known that there cannotbe a non-perturbative version of this result when d > C ∞ topology since our proof is KAM based. Thus, we exactly solvethe question Simon proposed in the Remarks and Historical Notes to [41,Section 10.16].(c) Going beyond the explicit formulation of Simon’s question, the curi-ous reader may wonder whether in addition to the pure point result from[45] and the absolute continuity result in Theorem 1.1 one can also exhibit Fayad-Krikorian [19] provide the C ∞ version of the KAM result underlying ourapproach. LONG LI, DAVID DAMANIK, AND QI ZHOU analytic (or smooth) quasi-periodic CMV matrices with purely singular con-tinuous spectrum. The answer is yes and all necessary tools to produce suchexamples (of extended analytic quasi-periodic CMV matrices with purelysingular continuous spectrum) already exist. We include a discussion to thiseffect in Appendix C for the sake of said curious reader.We remark that from the OPUC perspective, standard CMV matricesare the natural object and hence we are primarily interested in the spectralproperties of the matrices C x . However, from the perspective of the actualspectral analysis of these matrices, it is crucial to also consider the extendedCMV matrices E x since the underlying torus translation T ω : T d → T d , x x + ω is an invertible ergodic transformation , which enables us to rely on the pow-erful general theory that has been developed for extended CMV matriceswith Verblunsky coefficients generated by continuous sampling along the or-bit of such a transformation; compare [14, 18] and references therein. We willbe more specific below, but merely mention here that the spectrum can becharacterized in terms of the uniform hyperbolicity of the associated Szeg˝ococycles [14] and the absolutely continuous spectrum can be investigated viaKotani theory [18, 20, 22, 23, 34, 41] (and both of these fundamental toolsrequire the two-sided setting in their full-fledged version). Consequently,we will prove the desired absolute continuity of spectral measures both forthe standard CMV matrices C x and for the extended CMV matrices E x . Tofacilitate stating our results, let us mention now that given h , λ , and ω ,there is a compact set Σ ⊆ ∂ D such that for every x ∈ T d , we have(1.5) Σ = σ ess ( C x ) = σ ess ( E x ) = σ ( E x ) . The discrete spectrum of C x may (and usually will) be x -dependent, but it isknown that each connected component of ∂ D \ Σ contains at most one pointthat belongs to σ disc ( C x ). We will say more about this later in the paper.It is well known, and crucial to our present discussion, that the theoryof orthogonal polynomials on the unit circle shares many similarities withthe theory of one-dimensional discrete Schr¨odinger operators. This is nota coincidence as the latter embeds naturally into the theory of orthogonalpolynomials on the real line, and the unit circle and the real line stand out asthose subsets of the complex plane on which polynomials that are orthogonalwith respect to a non-trivial probability measure that is supported on oneof these two sets are known to obey a useful recursion (the Verblunskycoefficients are just the recursion coefficients in the case of the unit circle;compare (1.1)). As a result, there has been an extensive effort to workout OPUC analogs of results that had first been established for discreteSchr¨odinger operators. From this perspective, the result of Zhang mentionedabove [50] is an OPUC analog of results of Herman [29] and Sorets-Spencer[43] in the Schr¨odinger setting, and the Wang-Damanik result [45] is ananalog of a result of Bourgain-Goldstein [9]. The results of the present paper .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 5 are the OPUC analog of a Schr¨odinger operator result that has evolved overthe years with many important contributions. We will give more detailedpointers to the literature below, but mention here some of the milestones:Dinaburg-Sinai [16], Eliasson [17], Avila-Jitomirskaya [3], Avila [1].We wish to emphasize that working out the OPUC analog of a Schr¨odingerresult is not straightforward. There is no general mechanism that serves asa black box transforming an input into an output without further work.Rather, the plethora of known companion results exists primarily becausemany of the tools of spectral analysis on the Schr¨odinger side have beenshown to have analogs on the OPUC side, and using the latter allowed theauthors in question to obtain the desired OPUC results by following a similarproof strategy.In the case of the problem at hand, a crucial tool in the spectral analysisof the quasi-periodic Schr¨odinger case, namely Aubry duality, is not knownto exist in a similarly useful form in the OPUC setting. Aubry duality al-lows one, in the Schr¨odinger case, to relate the absolute continuity problemvia duality to a localization problem for the dual model and to exploit thepowerful methods that have been developed to establish such localizationstatements; we refer the reader to [1, 3, 10] for results of this kind. The ex-istence of absolutely continuous spectrum is well known to follow from theboundedness of the solutions of the difference equation associated with theoperator in question, and this boundedness can often be established in thequasi-periodic setting by describing these solutions with the help of suitablecocycles, and then conjugating these cocycles to SO(2 , R )-valued cocycles.The key issue is the pure absolute continuity as one typically obtains bound-edness only on a set of spectral parameters that has full Lebesgue measure,and hence the absence of singular spectrum still needs to be clarified. Tothis end we follow here an approach initially developed by Avila [1]. Inthis approach, one needs to prove that the set of spectral parameters withunbounded solutions has zero weight with respect to the spectral measure,which in turn relies on a measure estimate for the set of spectral parametersfor which the corresponding cocycles have a given growth rate. In partic-ular, one seeks to establish suitable almost reducibility results (where onecan conjugate into an arbitrarily small neighborhood of a constant). Theduality approach shows that (almost) reducibility for the initial model canbe derived from (almost) localization for the dual model. As we do nothave the duality approach at our disposal for the problem we study, wehave to look for direct methods of proving absolute continuity via (almost)reducibility. Instead we establish a quantitative almost reducibility resultdirectly with the help of recent advances by Cai-Chavaudret-You-Zhou [11]and Leguil-You-Zhao-Zhou [35] . We also mention that there is anotherproof of purely absolutely continuous spectrum for continuum Schr¨odinger This approach was first used to deal with the absolutely continuous spectrum ofSchr¨odinger operators with quasi-periodic-like potentials [46].
LONG LI, DAVID DAMANIK, AND QI ZHOU operators with small analytic quasi-periodic potentials by Eliasson [17] thatdoes not use duality. Compared with [17] our approach is more concise: wedo not need to complexify the energy and estimate the imaginary part ofthe Green function G ( E ± ig ; n, m ) (which corresponds to G ((1 ± δ ) e iζ ; n, m )in our case).Finally, we mention that in the Schr¨odinger context, the approach to thestudy of the spectral properties of a quasi-periodic operator that is basedon quantitative almost reducibility is very fruitful [1, 3, 6, 11, 17, 35] (seealso the nice survey of You [48] for more results). Our result is the firstrealization of this in the OPUC setting.The structure of the paper is as follows. After recalling some relevantparts of the general OPUC theory in Section 2, we prove a quantitative al-most reducibility result for analytic quasi-periodic SU(1 ,
1) cocycles (derivedfrom the work of Cai-Chavaudret-You-Zhou [11] and Leguil-You-Zhao-Zhou[35], which we briefly recall in Appendix A) in Section 3 and a lower boundfor the density of states measure in the quasi-periodic case we study in Sec-tion 4. The proof of Theorem 1.1 is given in Section 5. Finally, we discusshow to obtain a non-perturbative result in the case d = 1 in Appendix B andhow to obtain analytic quasi-periodic extended CMV matrices with singularcontinuous spectrum in Appendix C.2. Preliminaries
In this section we collect some material we will need in the subsequentsections. The results we describe here are well known, but they are includedfor the convenience of the reader.2.1.
The Standard Factorization of CMV Matrices.
Recall that, givena sequence { α n } n ∈ Z + ⊂ D = { z ∈ C : | z | < } , the standard (or half-line)CMV matrix takes the form C = α α ρ ρ ρ · · · ρ − α α − ρ α · · · α ρ − α α α ρ ρ ρ · · · ρ ρ − ρ α − α α − ρ α · · · α ρ − α α · · ·· · · · · · · · · · · · · · · · · · , where ρ j = p − | α j | . It is possible to factorize this matrix as follows, C = LM , where L = Θ Θ Θ . . . , M = Θ . . . .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 7 and Θ j = (cid:18) ¯ α j ρ j ρ j − α j (cid:19) . Similarly, the extended CMV matrix E = · · · · · · · · · · · · · · · · · · · · ·· · · − α α − α ρ ρ ρ · · ·· · · − ρ α − − α α − ρ α · · ·· · · α ρ − α α α ρ ρ ρ · · ·· · · ρ ρ − ρ α − α α − ρ α · · ·· · · α ρ − α α · · ·· · · · · · · · · · · · · · · · · · · · · can be written as E = L ′ M ′ , where L ′ = . . . Θ − Θ Θ . . . , M ′ = . . . Θ − Θ . . . . Maximal Spectral Measures.
By construction, the orthogonalitymeasure µ with respect to which the CMV matrix C is constructed is amaximal spectral measure for this matrix. In other words, for every ψ ∈ ℓ ( Z + ), the spectral measure corresponding to the pair ( C , ψ ) is absolutelycontinuous with respect to µ . In particular, C has spectral multiplicity one.An extended CMV matrix E , on the other hand, in general does not havespectral multiplicity one. However, the spectral multiplicity of E is at mosttwo and the spectral subspace corresponding to E and the pair of vectors δ , δ is all of ℓ ( Z ) [41, Theorem 10.16.5]. Thus, the sum of the spectralmeasures corresponding to ( E , δ ) and ( E , δ ) is maximal and will be denotedby Λ.For our later discussion let us keep in mind that in order to prove thatall spectral measures of a standard (resp., extended) CMV matrix C (resp., E ) are purely absolutely continuous on some set S ⊆ ∂ D , it suffices to showthat µ (resp., Λ) is purely absolutely continuous on S .Note also that for the quasi-periodic CMV matrices C x and E x we study,the maximal spectral measures of course depend on x ∈ T d and will bedenoted by µ x and Λ x , respectively.2.3. Transfer Matrices.
By normalizing the monic orthogonal polynomi-als Φ n ( z ) in L ( ∂ D , dµ ), we obtain the orthonormal polynomials ϕ n ( z ). LONG LI, DAVID DAMANIK, AND QI ZHOU
Their Szeg˝o duals ϕ ∗ n ( z ) are defined as before. The recursion (1.1) canthen be rephrased as follows,(2.1) (cid:18) ϕ n ( z ) ϕ ∗ n ( z ) (cid:19) = 1 ρ n (cid:18) z − ¯ α n − α n z (cid:19) (cid:18) ϕ n − ( z ) ϕ ∗ n − ( z ) (cid:19) , ϕ ( z ) = ϕ ∗ ( z ) = 1 , where the α n are the Verblunsky coefficients and ρ n is given by (1.2).The matrix ˜ S ( α, z ) = 1 p − | α | (cid:18) z − ¯ α − αz (cid:19) , which provides one step on the iteration of (2.1), is the called the Szeg˝ococycle map .In this paper, we find it convenient to work with the renormalized Szeg˝ococycle map (2.2) S ( α, z ) = z − ˜ S ( α, z ) ∈ SU(1 , . We need the following relations between the solutions of the generalizedeigenvalue equation of C x and the Szeg˝o polynomials. Suppose u, v are de-fined as C x u = zu, v = M u, that is, u is the solution of the generalized eigenvalue equation of C x and v isobtained by applying a unitary transformation on u . The vector (cid:18) u n v n (cid:19) thenobeys the Gesztesy-Zinchenko iterations. It relates to the Szeg˝o cocycle viathe following:(2.3) u n = z − n ϕ n ( z ) , u n +1 = z − ( n +1) ϕ ∗ n +1 ( z )(2.4) v n = z − n ϕ ∗ n ( z ) , v n +1 = z − n ϕ n +1 ( z )with (cid:18) u v (cid:19) = (cid:18) (cid:19) as boundary values, for all z ∈ C \{ } and n ≥ z ∈ ∂ D , we have | ϕ n | = | ϕ ∗ n | , which implies(2.5) | u n | = | ϕ n | , n ∈ N . For extended CMV matrices, there are very similar results. Let s, t be thesolutions of the following generalized eigenvalue equations E x s = zs, t = M ′ s. Then (cid:18) s n t n (cid:19) obeys the Gesztesy-Zinchenko iteration: for n ∈ Z ,(2.6) (cid:18) s n t n (cid:19) = T n (cid:18) s n − t n − (cid:19) , (cid:18) s t (cid:19) = (cid:18) (cid:19) .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 9 where(2.7) T n = ρ n (cid:18) − α n z − z − α n (cid:19) n is even,1 ρ n (cid:18) − α n − α n (cid:19) n is odd.Putting (2.1), (2.6), and (2.7) together for z ∈ ∂ D , a direct calculation gives(2.8) | s n | = | ϕ n | , n ∈ Z . Note that for n > ϕ − n ( z ) is obtained via(2.9) (cid:18) ϕ − n ( z ) ϕ ∗− n ( z ) (cid:19) = ˜ S ( α − n +1 , z ) − (cid:18) ϕ − n +1 ( z ) ϕ ∗− n +1 ( z ) (cid:19) , (cid:18) ϕ ( z ) ϕ ∗ ( z ) (cid:19) = (cid:18) (cid:19) . The ϕ − n ( z )’s are polynomials of z with degree n and ϕ ∗− n ( z ) = z − n ϕ n ( z ).This relation inspires us to study the Szeg˝o cocycle instead of the Gesztesy-Zinchenko cocycle directly.2.4. Dynamically Defined Verblunsky Coefficients.
In this subsectionwe emphasize that the Verblunsky coefficients we study are obtained by sam-pling along the orbits of a discrete-time dynamical system, and hence theyembed into the general theory of dynamically defined Verblunsky coeffi-cients. We explain the primary objects of interest in this scenario and someof the key results that are known to hold. However, for the sake of simplic-ity we do not make the general framework explicit, but rather introduce allquantities for the specific case at hand. We merely point out that many ofthe quantities and results below make sense in a more general setting.2.4.1.
The Density of States Measure.
Averaging the spectral measure cor-responding to the pair ( E x , δ ) with respect to the normalized Lebesguemeasure on T d , we obtain the density of states measure dk on ∂ D : Z g dk = Z T d h δ , g ( E x ) δ i dx for any g ∈ C ( ∂ D ). Note that by shift-invariance of the Lebesgue measureon T d , dk can alternatively be defined as one-half times the average of Λ x .Under suitable assumptions, the density of states measure can also beinterpreted as the density of zeros measure , which is the weak limit of thefinitely supported probability measures obtained by placing point masses atthe zeros of the orthogonal polynomials (according to their multiplicities).Note that the zeros of the orthogonal polynomials lie in the open unitdisk D and hence the existence of the density of zeros measure as a measureon the unit circle ∂ D is a non-trivial property that will not always hold. Fora very simple counterexample, note that the all zeros lie at the origin when We will sometimes view ∂ D as R / (2 π Z ). In particular, when we talk about the weightassigned by the density of states measure to an arc on the unit circle, we typically denotethat by k ( a, b ), where ( a, b ) is a real interval. the Verblunsky coefficients vanish, and hence the density of zeros measureexists in this case but is not a measure on the unit circle (rather it is thenormalized point measure at the origin). On the other hand, the density ofzeros exists as a measure on the unit circle when the function α defining theVerblunsky coefficients via (1.3) satisfies Z T d ln( | α ( x ) | ) dx > −∞ , compare [41, Theorem 10.5.19] and the general discussion in [41, Section 10.5].In view of (1.4), this sufficient condition holds true for every λ ∈ (0 , The Szeg˝o Cocycle, the Lyapunov Exponent, and the Rotation Num-ber. A quasi-periodic cocycle ( ω, A ) ∈ R d × C ( T d , SL(2 , C )) is a linear skewproduct: ( ω, A ) : T d × C → T d × C ( x, v ) ( x + ω, A ( x ) · v ) , where T d = R d / Z d and ω ∈ T d is rationally independent. For n ≥
1, theproducts are defined as A n ( x ) = A ( x + ( n − ω ) · · · A ( x ) , and A − n ( x ) = ( A n ( x − nω )) − , then we can define Lyapunov exponent as γ ( ω, A ) = lim n →∞ n Z ln k A n ( x ) k dx. Assume now that A ∈ C ( T , SL(2 , R )) is homotopic to the identity. Thenthere exist ψ : T d × T → R and u : T d × T → R + such that A ( x ) · (cid:18) cos 2 πy sin 2 πy (cid:19) = u ( x, y ) (cid:18) cos 2 π ( y + ψ ( x, y ))sin 2 π ( y + ψ ( x, y )) (cid:19) . The function ψ is called a lift of A . Let µ be any probability measure on T d × T which is invariant by the continuous map T : ( x, y ) ( x + ω, y + ψ ( x, y )),projecting over Lebesgue measure on the first coordinate (for instance, take µ as any accumulation point of n P n − k =0 T k ∗ ν , where ν is Lebesgue measureon T d × T ). Then the number ρ ( ω, A ) = Z ψ dµ mod Z does not depend on the choices of ψ and µ , and is called the fibered rotationnumber of ( ω, A ), see [29] and [33]. For any C ∈ SL(2 , R ) , it is immediatefrom the definition that we have the following:(2.10) | ρ ( ω, A ) − ρ ( ω, C ) | ≤ k A ( x ) − C k C . .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 11 Once we have the definition of the fibered rotation number for SL(2 , R )cocycles, we can parlay this concept to SU(1 ,
1) cocycles, since M = 11 + i (cid:18) − i i (cid:19) induces an isomorphism between SL(1 ,
1) and SL(2 , R ): M − SU(1 , M = SL(2 , R ) . The
Szeg˝o cocycle is defined by letting α n ( x ) = α ( x + ( n − ω ) and A z = S ( α, z ) (with S ( α, z ) as in (2.2)). Denote this cocycle by ( ω, S ( α, z )).The rotation number ρ ( z ) = ρ ( ω, S ( α, z )) of the Szeg˝o cocycle and theDOS measure dk associated with the family E x are related by the followingformula [41, Theorem 8.3.3]:(2.11) 2 ρ ( e iζ ) = k (0 , ζ ) , ζ ∈ [0 , π ) . We also denote the
Lyapunov exponent of the Szeg˝o cocycle and the renor-malized Szeg˝o cocycle by˜ γ ( z ) = γ ( ω, ˜ S ( α, z )) , γ ( z ) = γ ( ω, S ( α, z )) , respectively. It follows readily from (2.2) that(2.12) γ ( z ) = ˜ γ ( z ) −
12 log | z | . Note in particular that γ ( z ) and ˜ γ ( z ) coincide for z ∈ ∂ D .The Lyapunov exponent and the DOS measure are related by the wellknown Thouless formula (cf. [41, Section 10.5]):(2.13) ˜ γ ( z ) = − ln ρ ∞ + Z ln | − ze − iθ | dk ( θ ) , where ρ ∞ = exp (cid:18) Z log(1 − | α ( x ) | ) dx (cid:19) . Uniform Hyperbolicity and the Set Σ . We say that a cocycle ( ω, A )is uniformly hyperbolic if there exist constants c, γ > CP = E s ( x ) ⊕ E u ( x ), x ∈ T d , such that(1) A ( x ) E ∗ ( x ) = E ∗ ( x + ω ) for ∗ = s, u .(2) For n ∈ Z + , x ∈ X, ξ u ∈ E u ( x ) , ξ s ∈ E s ( x ), k A n ( x ) ξ s k ≤ ce − γn k ξ s k , k A − n ( x ) ξ u k ≤ ce − γn k ξ u k . We denote by
U H the set of all uniformly hyperbolic cocycles.The following result from [14] relates the spectrum of extended CMVmatrices to their Szeg˝o cocycles:
Theorem 2.1.
There is a compact set Σ ⊂ ∂ D with σ ( E x ) = Σ for every x ∈ T d . Moreover, this uniform spectrum Σ is equal to ∂ D \ U , where U = { z ∈ ∂ D : ( ω, ˜ S ( α, z )) ∈ U H} . We emphasize two important points. The spectrum in the two-sided caseis independent of x and it is described by an explicit dynamical property ofthe associated Szeg˝o cocycles. For this reason, it is beneficial to consider thetwo-sided case in the dynamically defined situation (assuming the underlyingdynamical system is invertible) even if one is ultimately interested in thespectral analysis of the one-sided case. Namely, the two-sided spectrum Σ isdetermined via the above characterization in terms of the absence of uniformhyperbolicity, and this set then in turn serves as the essential spectrum ofthe one-sided CMV matrix C x , which is x -independent as pointed out above.Moreover, the x -dependent discrete spectrum of C x is quite tame: there isat most one point in each gap of Σ.3. Quantitative Almost Reducibility
Our proof is based on quantitative almost reducibility results for SU(1 , ω, A ) is analytically conjugate to ( ω, A ) if thereexists B ∈ C ω (2 T d , SU(1 , B ( · + ω ) A ( · ) B − ( · ) = A ( · ) . The cocycle ( ω, A ) is said to be almost reducible if the closure of its analyticconjugations contains a constant.For any sufficiently small ǫ > r >
0, let us define the followingsequences: ǫ j = ǫ j , r j = r j , N j = 4 j +1 ln ǫ − r . Then we have the following quantitative almost reducibility results, whichare based on the modified KAM scheme developed in [11, 35] (see Proposi-tion A.1 in Appendix A for the precise statement we employ).
Proposition 3.1.
Assume that κ, τ, r > and ω ∈ DC( κ, τ ) . Let S ∈ SU(1 , , f ∈ C ωr ( T d , su(1 , with k f k r ≤ ǫ ≤ D k S k C (cid:16) r (cid:17) C τ , where D = D ( κ, τ ) and C is a numerical constant. Then for any j ≥ ,there exists B j ∈ C ωr j (2 T d , SU(1 , such that B j ( x + ω )( S e f ( x ) ) B − j ( x ) = S j e f j ( x ) , where k f j ( x ) k r j ≤ ǫ j and B j satisfies k B j k ≤ ǫ − j − , (3.1) | deg B j | ≤ N j − . (3.2)(a) For any < | m | < N j − , we denote Λ m ( j ) = { z ∈ Σ : k ρ ( ω, S j − ) − h m, ω ik R / Z < ǫ j − } . (3.3) .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 13 Then if z ∈ Λ m ( j ) , we have the following precise expression: S j = exp (cid:18) it j v j ¯ v j − it j (cid:19) , where t j ∈ R , v j ∈ C , and | t j | ≤ ǫ j − , | v j | ≤ ǫ j − . (b) Moreover, there always exist unitary matrices U j ∈ SL(2 , C ) such that (3.4) U j S j e f j ( x ) U − j = (cid:18) e πiρ j c j e − πiρ j (cid:19) + F j ( x ) where ρ j ∈ R ∪ i R , with estimates k F j k r j ≤ ǫ j , and (3.5) k B j k | c j | ≤ k S k . Proof.
We will prove the result by iteration.
First step:
By our choice of ǫ , we can apply Proposition A.1 once to( ω, S e f ( x ) ) and obtain the following:There exists B ∈ C ωr (2 T d , SU(1 , B ( x + ω ) S e f ( x ) B − ( x ) = S e f ( x ) with the following estimates: k f k r ≤ ǫ , k B k ≤ ǫ − , | deg B | ≤ N . Inductive step:
Suppose we have already completed the j -th step andare at the ( j + 1)-th step. That is, there exists B j ∈ C ωr j (2 T d , SU(1 , B j ( x + ω ) S e f ( x ) B − j ( x ) = S j e f j ( x ) with estimates k f j k r j ≤ ǫ j , k B j k ≤ ǫ − j − , | deg B j | ≤ N j − . Now we consider the ( j + 1)-th step. By our choice of ǫ , one can check that(3.6) ǫ j ≤ D k S j k C ( r j − r j +1 ) C τ . Indeed, ǫ j on the left side of (3.6) decays super-exponentially with j , while( r j − r j +1 ) C τ on the right side decays exponentially with j . Note that (3.6)implies that Proposition A.1 can be applied iteratively, and hence we canconstruct¯ B j +1 ∈ C ωr j +1 (2 T d , SU(1 , , S j +1 ∈ SU(1 , , f j +1 ∈ C ωr j +1 ( T d , su(1 , B j +1 ( x + ω ) S j e f j ( x ) ¯ B − j +1 ( x ) = S j +1 e f j +1 ( x ) . Let B j +1 = ¯ B j +1 B j , then we have B j +1 ( x + ω ) S e f ( x ) B − j +1 ( x ) = S j +1 e f j +1 ( x ) . In order to verify (3.1) and (3.2) for B j +1 , we need to distinguish the fol-lowing two cases: Non-resonant case:
If for any n ∈ Z d with 0 < | n | ≤ N j we have k ρ ( ω, S j ) − h n, ω ik R / Z ≥ ǫ j , then by the non-resonant case of Proposition A.1, (cid:13)(cid:13) ¯ B j +1 − id (cid:13)(cid:13) r j +1 ≤ ǫ j , k f j +1 k r j +1 ≤ ǫ j +1 . In this case we have deg B j +1 = deg B j , since ¯ B j +1 is homotopic to theidentity. Therefore, k B j +1 k ≤ (1 + ǫ j ) ǫ − j − ≤ ǫ − j , | deg B j +1 | = | deg B j | ≤ N j − ≤ N j . These verify (3.1) and (3.2) for B j +1 in this case. Resonant case: If z ∈ Λ m ( j + 1) for some m ∈ Z d with 0 < | m | ≤ N j ,then by the resonant case of Proposition A.1, we have deg ¯ B j +1 = m and k ¯ B j +1 k ≤ C | m | τ ≤ ǫ − j , k f j +1 k r j +1 ≤ ǫ j e − r j +1 ǫ − τj ≤ ǫ j +1 . Moreover, we can write S j +1 = exp (cid:18) it j +1 v j +1 ¯ v j +1 − it j +1 (cid:19) , where t j +1 ∈ R , v j +1 ∈ C with | t j +1 | ≤ ǫ j , | v j +1 | ≤ ǫ j . Therefore, k B j +1 k ≤ ǫ − j − ǫ − j ≤ ǫ − j , | deg B j +1 | ≤ N j − + N j ≤ N j . By induction we have proven (3.1) and (3.2) for each j ≥
1. As a conse-quence of the resonant case, the statement of ( a ) was verified. We are leftto prove the statement ( b ).From the above iteration, we know after each resonant step j , one canwrite S j = exp S ′′ j with k S ′′ j k ≤ ǫ j − . Let U j ∈ SL(2 , C ) such that (3.4)holds, then(3.7) | c j | ≤ k S ′′ j k ≤ ǫ j − . Suppose there are two resonance sites n j i , n j i +1 , which happen at theKAM steps j i +1 , j i +1 +1, respectively. Then we want to show that | n j i +1 | ≥ ǫ − τ j i | n j | . By the resonance condition of the ( j i +1 + 1)-th step, | ρ j i +1 − h n ji +1 ,ω i | ≤ ǫ j i +1 , hence | ρ j i +1 | > κ | n ji +1 | τ . On the other hand, accordingto Proposition A.1, after the ( j i + 1)-th step, | ρ j i +1 | ≤ ǫ j i . Thus we have | n j i +1 | ≥ ǫ − τ j i | n j i | . This implies that the resonant steps are actually very far from each other. .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 15
Since the steps between j i + 1 and j i +1 + 1 are all non-resonant, accordingto the non-resonant case of Proposition A.1, we have the following: For j i + 1 < j ≤ j i +1 ,(3.8) k S j − S j − k ≤ ǫ j − . Since the ( j i + 1)-th step is resonant, we have S j i +1 = exp (cid:18) it j i +1 v j i +1 ¯ v j i +1 − it j i +1 (cid:19) with | t j i +1 | ≤ ǫ j i and | v j i +1 | ≤ ǫ j i . There exists a unitary matrix U j i +1 ∈ SL(2 , C ) such that U j i +1 S j i +1 e f ji +1( x ) U − j i +1 = (cid:18) e πiρ ji +1 c j i +1 e − πiρ ji +1 (cid:19) + F j i +1 ( x ) , where | c j i +1 | ≤ k S ′′ j i +1 k ≤ ǫ j i and k F j i +1 k r ji +1 ≤ ǫ j i +1 . Let S j = exp S ′′ j .By (3.8), we have the following: k S ′′ j k ≤ ǫ j i + j X k = j i +2 k S k − S k − k ≤ ǫ j i . In this case, we have(3.9) | c j | ≤ k S ′′ j k ≤ ǫ j i . Since k B j i +1 k ≤ ǫ − j i and all steps between j i + 1 and j (including j ) arenon-resonant, we have(3.10) k B j k ≤ Π jk = j i +1 (1 + ǫ k ) ǫ − j i ≤ ǫ − j i . From the above argument and (3.1),(3.7), we can see that after eachresonant step j , we have k B j k | c j | ≤ ǫ j − . While after each non-resonant step j , we are able to utilize the estimates ofthe last resonant step j i + 1 if it exists. By (3.9) and (3.10) we have k B j k | c j | ≤ ǫ j i . However, it is possible that no resonant steps happened within the first j steps. In this case, each step is non-resonant and thus we can use theestimate k ¯ B i k ≤ ǫ i − for each i ≤ j and obtain k B j k ≤ Π i ≥ (1 + ǫ i ) ≤ . Since S j − S = P ji =1 ( S i − S i − ) and (3.8), we have k S j k ≤ k S k + 2 ǫ ≤ k S k . This finishes the proof. (cid:3) Let(3.11) K j = [ < | m |≤ N j − Λ m ( j )with Λ m ( j ) from (3.3). Then for any z ∈ K j , we have the following estimatefor the growth of the transfer matrix A sz : Corollary 3.2.
For z ∈ K j , we have sup ≤ s ≤ Cǫ − j − k A sz k ≤ Cǫ − j − . Proof.
Since z ∈ K j , by Proposition 3.1 there exists B j such that B j ( x + ω ) S e f ( x ) B − j ( x ) = S j e f j ( x ) , where S j = exp (cid:18) it j ¯ v j v j − it j (cid:19) and | t j | ≤ ǫ j − , | v j | ≤ ǫ j − , k f j k ≤ ǫ j . Moreover, we have k S j k ≤ ǫ j − . Since we have k B j k ≤ ǫ − j − by (3.1),this implies that sup ≤ s ≤ Cǫ − j − k A sz k ≤ C k B j k ≤ Cǫ − j − , concluding the proof. (cid:3) We also have the following:
Lemma 3.3.
For z ∈ K j , there exists n j ∈ Z d with | n j | ≤ N j − such that k ρ ( ω, S e f ( x ) ) − h n j , ω ik R / Z ≤ ǫ j − . Proof.
For z ∈ K j , we have k ρ j − − h n ∗ , ω ik R / Z ≤ ǫ j − for some n ∗ ∈ Z d with 0 < | n ∗ | ≤ N j − . Since B j − ( · ) conjugates ( ω, S e f ( · ) )into ( ω, S j − e f j − ( · ) ), we have2 ρ ( ω, S e f ( x ) ) + h deg B j − , ω i = 2 ρ ( ω, S j − e f j − ( x ) ) . By (2.10) and k f j − k ≤ ǫ j − , we have | ρ ( ω, S j − e f j − ( x ) ) − ρ j − | ≤ cǫ j − . These together give | ρ ( ω, S e f ( x ) ) + h deg B j − , ω i − h n ∗ , ω i| ≤ ǫ j − . .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 17 Let n j = − deg B j − + n ∗ . Then k ρ ( ω, S e f ( x ) ) − h n j , ω ik R / Z ≤ ǫ j − . Moreover, by (3.2) of Proposition 3 . , we have | deg B j − | ≤ N j − , and thisimplies that | n j | ≤ N j − + N j − ≤ N j − , as claimed. (cid:3) A Lower Bound for the DOS
In this section we consider CMV matrices with analytic quasi-periodicVerblunsky coefficients of the form α n ( x ) = α ( x + ( n − ω ) , n ∈ Z , x ∈ T d , ω ∈ DC( κ, τ ) , where α ( x ) = λe πih ( x ) , h ( x ) ∈ C ω ( T d , R ) , λ ∈ (0 , . Since the x -independent spectrum Σ of E x is a subset of ∂ D , we will use thenotation e iζ ∈ Σ to make this explicit.We will show that if λ is small enough, then we can establish (see Lemma 4.5below) a lower bound for the DOS measure for C x of the form(4.1) k ( ζ − ǫ, ζ + ǫ ) ≥ cǫ , e iζ ∈ Σ , which will be a key ingredient in the subsequent considerations.If | λ | is small enough, then S ( α, z ) is close to a constant, and henceemploying Proposition 3.1 we will be able to show (see Corollary 4.3 below)that the DOS measure is -H¨older continuous on Σ. Once this has beenaccomplished, (4.1) follows from the well-known Thouless formula.Denoting ρ = p − | α | = √ − λ , we have the following decomposition S ( α, z ) = z z − ! ρ − (cid:18) − αz − − αz (cid:19) . Let S ( α, z ) = z z − ! and P ( x ) = ρ − (cid:18) − λe − πih ( x ) z − − λe πih ( x ) z (cid:19) . Since P ( x ) − id = 1 ρ (cid:18) − ρ − λe πih ( x ) z − − λe πih ( x ) z − ρ (cid:19) for z ∈ ∂ D , we have k P ( x ) − id k r ≤ | λ | e π | h | r . Since the tangent space T id SU(1 ,
1) is the algebra su(1 , id ∈ SU(1 ,
1) and that of 0 ∈ su(1 , λ is sufficiently small,we have f ( x ) ∈ C ωr ( T d , su(1 , e f ( x ) = P ( x ), and hence we have S ( α, z ) = S ( α, z ) e f ( x ) . The smallness of λ can be viewed as the smallnessof k f k r . If we further assume that λ satisfies the condition(4.2) | λ | ≤ D e π | h | r (cid:16) r (cid:17) C τ , where the constants C , D are consistent with Proposition A.1, then wecan apply Proposition 3.1 to the system ( ω, S e f ( x ) ), and we obtain thefollowing consequences. Lemma 4.1. If λ obeys (4.2) , then for any e iζ ∈ Σ and ǫ > , we have (4.3) γ ( e iζ ) = 0 and (4.4) γ ((1 + ǫ ) e iζ ) ≤ Cǫ . Proof.
To prove (4.3), it suffices to show that if e iζ ∈ Σ, the cocycle growsat most linearly, that is,(4.5) k A ne iζ k ≤ Cn, n ≥ . We can apply Proposition 3.1 and distinguish two cases:
Case 1:
If ( ω, S e f ( x ) ) is reducible, then there exists B ∈ C ω (2 T d , SU(1 , B ( x + ω ) S e f ( x ) B − ( x ) = S. Since e iζ ∈ Σ, by Theorem 2.1, the cocycle ( ω, S e f ( x ) ) is not uniformlyhyperbolic, and hence we have S = (cid:18) e πiρ c e − πiρ (cid:19) with ρ ∈ R . This implies that k A ne iζ k ≤ k B k (1 + cn ) ≤ Cn for n ≥ Case 2:
If ( ω, S e f ( x ) ) is not reducible but almost reducible, we need thefollowing claim in order to describe the growth of the cocycle more precisely: Claim 1.
Suppose e iζ ∈ Σ , then for each j > , there exists ˜ B j ( x ) such that (4.6) ˜ B j ( x + ω )( S e f ( x ) ) ˜ B − j ( x ) = (cid:18) e πiρ j c j e − πiρ j (cid:19) + ˜ F j ( x ) , with ρ j ∈ R , k ˜ B j k ≤ ǫ − j − , k ˜ F j k ≤ ǫ j , k ˜ B j k | c j | ≤ k S k .Proof. By Proposition 3.1, we obtain B j ( x ) such that B j ( x + ω ) S e f ( x ) B − j ( x ) = (cid:18) e πiρ j c j e − πiρ j (cid:19) + F j ( x ) , .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 19 with ρ j ∈ R ∪ i R , k B j k ≤ ǫ − j − , k F j k ≤ ǫ j and k B j k | c j | ≤ k S k . If ρ j ∈ R , then let ˜ B j ( x ) = B j ( x ) and ˜ F j ( x ) = F j ( x ), the claim followsimmediately. Assume that ρ j ∈ i R with | ρ j | > ǫ j and let Q j = (cid:18) q j q − j (cid:19) , where q j = k B j k ǫ j . Then we have Q j (cid:20)(cid:18) e πiρ j c j e − πiρ j (cid:19) + F j ( x ) (cid:21) Q − j = (cid:18) e πiρ j e − πiρ j (cid:19) + F ′ j ( x ) , where F ′ j ( x ) = QF j Q − + (cid:18) c j q j (cid:19) . Since | c j | q j = | c j |k B j k ǫ j and k Q j F j ( x ) Q − j k ≤ ǫ j , we have k F ′ j k ≤ Cǫ j .We want to show that this implies the system is uniformly hyperbolic. Givena non-zero vector ( a, b ) T ∈ R with | a | ≥ | b | , let (cid:18) a ′ b ′ (cid:19) = (cid:20)(cid:18) e πiρ j e − πiρ j (cid:19) + F ′ j ( x ) (cid:21) (cid:18) ab (cid:19) = (cid:18) e πiρ j ae − πiρ j b (cid:19) + F ′ j ( x ) (cid:18) ab (cid:19) . Without loss of generality, assume 2 πiρ j >
0. This implies | a ′ | ≥ ( e πiρ j − k F ′ j ( x ) k ) | a || b ′ | ≤ e − πiρ j | b | + 2 k F ′ j ( x ) k | a | . Therefore, | a ′ | − | b ′ | ≥ (4 πiρ j − Cǫ j ) | a | ≥
0. By the cone field criterion(compare, e.g., [47]), this implies the uniform hyperbolicity of ( ω, S ( α, e iζ )),which conflicts with our assumption that e iζ ∈ Σ, again by Theorem 2.1.So we have | ρ j | ≤ ǫ j and we put it into the perturbation to obtain thefollowing:˜ B j ( x + ω )( S e f ( x ) ) ˜ B − j ( x ) = (cid:18) c j (cid:19) + ˜ F j ( x ) , k ˜ F j k ≤ ǫ j , where ˜ B j ( x ) = B j ( x ). This reduces to the case ρ j = 0 and Claim 1 isproved, as we indeed have ρ j ∈ R . (cid:3) In order to control the growth of the cocycle, we need the following lemmaproved by Avila-Fayad-Krikorian [2]:
Lemma 4.2.
We have that M l (id + ξ l ) · · · M (id + ξ ) = M ( l ) (id + ξ ( l ) ) , where M ( l ) = M l · · · M and k ξ ( l ) k e P lk =0 k M ( k ) k k ξ k k − . By the result of Claim 1, let M k = S j = (cid:18) e πiρ j c j e − πiρ j (cid:19) , ξ k = S − j ˜ F j ( x + kω ), and apply Lemma 4.2 to obtain A ne iζ = ˜ B j ( x + nω ) S nj ( id + ξ ( n ) ) ˜ B − j ( x ) , where k ξ ( n ) k ≤ e P nk =1 k S kj k k ˜ F j k −
1. Since ρ j ∈ R , we have k S kj k ≤ k | c j | .These together with k ˜ F j k ≤ ǫ j give k A ne iζ k ≤ k ˜ B j k (1 + n | c j | ) e P nk =1 (1+ k | c j | ) ǫ j ≤ k ˜ B j k (1 + n | c j | ) e n ǫ j . Therefore, by (3.5) and the fact that k S k ≤
0, there exists j such that ǫ ∈ I j . To prove (4.4), we need the following result: Claim 2.
There exists W : 2 T d → SU(1 , analytic with k W k ≤ Cǫ − such that Q ( x ) = W ( x + ω ) S e f ( x ) W − ( x ) satisfies the estimate (4.8) k Q k ≤ Cǫ . Proof.
Let ˜ B j ( x ) , ˜ F j ( x ) be as above in Claim 1, and let D = (cid:18) d d − (cid:19) , where d = (cid:13)(cid:13)(cid:13) ˜ B j (cid:13)(cid:13)(cid:13) ǫ , let W ( x ) = D ˜ B j ( x ). Since ǫ ∈ I j , we have d ≤ k W k ≤ Cǫ − . By (4.6) we have W ( x + ω ) S e f ( x ) W − ( x ) = (cid:18) e πiρ j e − πiρ j (cid:19) + ˜ F ′ j ( x ) , where ˜ F ′ j ( x ) = (cid:13)(cid:13)(cid:13) ˜ B j (cid:13)(cid:13)(cid:13) ǫ c j ! + D ˜ F j ( x ) D − , .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 21 since k ˜ B j k | c j | ≤ k S k according to (3.5), we have k ˜ F ′ j k ≤ Cǫ and ρ j ∈ R .Then, with Q ( x ) = W ( x + ω )( S e f ( x ) ) W − ( x ), (4.8) follows immediately.This completes the proof of Claim 2. (cid:3) Let A = S ( α, e iζ ) , B = S ( α, (1 + ǫ ) e iζ ), then k A − B k ≤ ǫ. By the aboveargument, there exists W : 2 T d → SU(1 ,
1) such that C = W ( x + ω ) BW − ( x ) = W ( x + ω ) AW − ( x ) + W ( x + ω )( B − A ) W − ( x ) . Thus we have k C k ≤ k Q k + Cǫ , by Claim 2 and our choice of W , k C k ≤ Cǫ . It follows that γ ((1 + ǫ ) e iζ ) ≤ ln k C k ≤ Cǫ . This concludes theproof of Lemma 4.1. (cid:3) As a direct corollary, we have the following:
Corollary 4.3.
The DOS measure is -H¨older continuous, that is for any e iζ ∈ ∂ D and ǫ > small enough, we have (4.9) k ( ζ − ǫ, ζ + ǫ ) ≤ Cǫ . Proof.
Since the DOS measure is supported by Σ, we can limit our attentionto the case e iζ ∈ Σ. By the Thouless formula (2.13), we have˜ γ ( re iζ ) = − ln ρ ∞ + Z ln | − re i ( ζ − θ ) | dk ( θ ) . For any ǫ which is small enough, using the fact ˜ γ ( e iζ ) = γ ( e iζ ) = 0 byLemma 4 .
1, we have˜ γ ((1 + ǫ ) e iζ ) = ˜ γ ((1 + ǫ ) e iζ ) − ˜ γ ( e iζ )= Z ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 + ǫ ) e i ( ζ − θ ) − e i ( ζ − θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dk ( θ )= 12 Z ln (1 − (1 + ǫ ) e i ( ζ − θ ) )(1 − (1 + ǫ ) e − i ( ζ − θ ) )(1 − e i ( ζ − θ ) )(1 − e − i ( ζ − θ ) ) ! dk ( θ )= 12 Z ln (cid:18) ǫ + ǫ − ζ − θ ) (cid:19) dk ( θ ) . Therefore,˜ γ ((1 + ǫ ) e iζ ) ≥ Z ζ + ǫζ − ǫ ln (cid:18) ǫ + ǫ − ζ − θ ) (cid:19) dk ( θ ) ≥
12 ln (cid:18) ǫ + ǫ − ǫ (cid:19) k ( ζ − ǫ, ζ + ǫ ) , take ǫ sufficiently small, this implies˜ γ ((1 + ǫ ) e iζ ) ≥ ln 22 k ( ζ − ǫ, ζ + ǫ ) . Thus, by (2.12) and (4.4), we have k ( ζ − ǫ, ζ + ǫ ) ≤ γ ((1 + ǫ ) e iζ ) + 12 log(1 + ǫ )) ≤ Cǫ for any e iζ ∈ Σ, concluding the proof. (cid:3)
We also need the following general property of the
Lyapunov exponent .One can find this result in [40, Proposition 8.1.10].
Lemma 4.4.
Let e iζ ∈ ∂ D , then for any δ ≥ , we have ˜ γ ((1 + δ ) e iζ ) ≥ ln(1 + δ ) . With this result, we can prove the following:
Lemma 4.5.
For e iζ ∈ Σ and sufficiently small ǫ > , we have k ( ζ − ǫ, ζ + ǫ ) ≥ cǫ . Proof.
Let δ = cǫ . For e iζ ∈ Σ, we have ˜ γ ( e iζ ) = 0 by Lemma 4.1. Thus,by a calculation from the proof of Corollary 4.3, we have(4.10) ˜ γ ((1 + δ ) e iζ ) = 12 Z ln (cid:18) δ + δ − θ − ζ ) (cid:19) dk ( θ ) . Partition the integration region into {| θ − ζ | ≥ π } , { ǫ ≤ | θ − ζ | < π } , { ǫ ≤ | θ − ζ | < ǫ } , {| θ − ζ | < ǫ } , so that with the corresponding partition ofthe RHS of (4.10) into the four terms I , I , I , I , we have ˜ γ ((1 + δ ) e iζ ) = I + I + I + I .As I ≤ Z ln (cid:18) δ + δ − π (cid:19) dk ( θ ) , for δ sufficiently small, we have I ≤ δ .Furthermore, we have I = X k ≥ Z ǫ k > | θ − ζ |≥ ǫ k +1 ln (cid:18) δ + δ − θ − ζ ) (cid:19) dk ( θ ) ≤ X k ≥ ǫ k ln (cid:18) δ ǫ k +2 (cid:19) ≤ X k ≥ ǫ k ln(1 + 2 c ǫ − k +1 ) ≤ ǫ . With m = [ln πǫ − ], we have I ≤ m X k =0 Z π e − k − ≤| θ − ζ | < π e − k ln (cid:18) δ + δ − θ − ζ ) (cid:19) dk ( θ ′ ) . .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 23 Suppose k > δ > δ − πe − k − for k ≤ k , and δ ≤ δ − πe − k − for k < k ≤ m. Then, I ≤ δ k X k =0 π e − k + 2 m X k = k +1 π e − k δ − π e − k − ≤ δ π
100 11 − e − + 2 δ m X k =1 π
100 10000 e k +2 π ≤ δ
10 + 2 × δ e π − − e m − e < δ
10 + Cδ e m ≤ δ
10 +
Ccδ.
It follows that I ≥ ˜ γ ((1 + δ ) e iζ ) − δ − Ccδ.
By Lemma 4.4 we have ˜ γ ((1 + δ ) e iζ ) ≥ ln(1 + δ ) . Since the constant c above is consistent with our choice of δ , we can thereforeshrink it such that I ≥ δ . Since I ≤ Ck ( ζ − ǫ, ζ + ǫ ) ln ǫ − , the resultfollows. (cid:3) Proof of the Main Theorem
Our aim is to prove Theorem 1.1 in the present section. As this theo-rem makes statements both about the discrete spectrum and the essentialspectrum, we divide the discussion into two parts and begin with the easierissue of understanding the discrete spectrum. In fact, everything we needwill follow quickly from the general theory. The work from the previoussections then comes into play when we turn our attention to the essentialspectrum.5.1.
Discrete Eigenvalues.
In this subsection we begin with a discussionof the discrete spectrum. As it is well known that for every x ∈ T d , σ ( E x ) = σ ess ( E x ) = Σ, and hence the discrete spectrum of E x is empty, we can limitour attention to the discrete spectrum of C x . Theorem 5.1.
For every x ∈ T d , σ ess ( C x ) = Σ , and there is at most oneeigenvalue of C x in each connected component of ∂ D \ Σ .Proof. By [40, Theorem 4.5.2], E x − Q = C x L K x , where Q is trace class and C x , K x are half-line CMV matrices, acting on ℓ ( Z + ) and ℓ ( {· · · , − , − } ),respectively. Thus, σ ess ( E x ) = σ ess ( C x ) ∪ σ ess ( K x ) (cf. [13, Theorem 3]).Moreover, σ ess ( C x ) = σ ess ( K x ) as the translation T x = x + ω is minimalwhenever ω is rationally independent. Combining this with σ ess ( E x ) = Σ,we obtain σ ess ( C x ) = Σ.According to [41, Theorem 10.16.3], in each connected component of ∂ D \ Σ, C x has at most one eigenvalue. Note that [41, Theorem 10.16.3]only gives this result for almost every x ∈ T d , as in the general dynamicallydefined setting, the condition supp( dk ) = σ ( E x ) only holds for almost every x ∈ T d . But in our situation we have σ ess ( C x ) = Σ for each x ∈ T d , andhence via the same line of reasoning, the statement can be derived for each x ∈ T d . (cid:3) This completes the proof of the statements in Theorem 1.1 concerningspectral parameters in ∂ D \ Σ. It remains to prove the pure absolute conti-nuity of all spectral measures, of both C x and E x for arbitrary x ∈ T d , insidethe common essential spectrum Σ.5.2. Bounded Eigenfunctions and Absolutely Continuous Spectrum.
In order to prepare for our proof of the desired absolute continuity statement,which will be given in the next subsection, we discuss in this subsection howto connect the problem to the cocycle norm estimates we have obtained sofar.Given φ ∈ ∂ D , let µ φ be a nontrivial probability measure on ∂ D such that α n ( dµ φ ) = φα n ( dµ ). Let F ( z ) = Z e iθ + ze iθ − z dµ and F φ ( z ) = Z e iθ + ze iθ − z dµ φ be the Carath´eodory functions of µ, µ φ , respectively. Then [40, Theorem 3.2.14]implies(5.1) F φ = (1 − φ ) + (1 + φ ) F (1 + φ ) + (1 − φ ) F .
Let ϕ φn ( z ) , ψ φn ( z ) be the orthogonal polynomials of dµ φ . Then according to[40, Proposition 3.2.1] we have ϕ φn ( z )¯ φ ( ϕ φn ) ∗ ( z ) ! = 1 p − | α n | (cid:18) z − ¯ α n − α n z (cid:19) ϕ φn − ( z )¯ φ ( ϕ φn − ) ∗ ( z ) ! .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 25 with the initial condition (cid:18) φ (cid:19) . Similarly, ψ φn ( z ) obeys ψ φn ( z ) − ¯ φ ( ψ φn ) ∗ ( z ) ! = 1 p − | α n | (cid:18) z − ¯ α n − α n z (cid:19) ψ φn − ( z ) − ¯ φ ( ψ φn − ) ∗ ( z ) ! with initial condition (cid:18) − ¯ φ (cid:19) .The following result is a consequence of the Jitomirskaya-Last inequality,which was first proven in the Schr¨odinger case [32] and then worked out inthe OPUC setting [41]. Its purpose is to connect the growth of the cocyclewith the Carath´eodory function of the corresponding spectral measure. Proposition 5.2.
For any e iζ ∈ ∂ D and < ǫ < , we have the following: (5.2) sup φ ∈ ∂ D | F φ ((1 − ǫ ) e iζ ) | ≤ C sup ≤ s ≤ cǫ − (cid:13)(cid:13) A se iζ (cid:13)(cid:13) . Since we did not find this statement in the exact form we need, we in-clude an explanation of how to derive it from the OPUC version of theJitomirskaya-Last inequality for the convenience of the reader.For l ∈ (0 , ∞ ), define k a k l = [ l ] X j =0 | a j | + ( l − [ l ]) | a [ l ]+1 | . Let ϕ φ ( z ) = { ϕ φ ( z ) , ϕ φ ( z ) , · · · } , ψ φ ( z ) = { ψ φ ( z ) , ψ φ ( z ) , · · · } . Then it isknown that (cid:13)(cid:13) ϕ φ (cid:13)(cid:13) l (cid:13)(cid:13) ψ φ (cid:13)(cid:13) l runs from 1 to ∞ in a strictly monotone way as l runs from 0 to ∞ . The CMV version of the Jitomirskaya-Last inequalitynow reads as follows (cf. [41, Theorem 10.8.2]): Lemma 5.3.
For e iζ ∈ ∂ D and r ∈ [0 , , define l ( r ) to be the uniquesolution of (5.3) (1 − r ) (cid:13)(cid:13)(cid:13) ϕ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( r ) (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( r ) = √ . Then (5.4) A − " (cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13) l ( r ) k ϕ φ ( e iζ ) k l ( r ) ≤ | F φ ( re iζ ) | ≤ A " (cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13) l ( r ) k ϕ φ ( e iζ ) k l ( r ) , where A is a universal constant. With the help of this lemma we can now prove Proposition 5.2.
Proof.
Applying Lemma 5.3 with r = 1 − ǫ , (5.3) turns into (cid:13)(cid:13)(cid:13) ϕ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) = √ ǫ , the pair of inequalities in (5.4) becomes(5.5) ( √ A ) − ǫ (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) ≤ | F φ ((1 − ǫ ) e iζ ) | ≤ A √ ǫ (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) . Since ϕ φn ( e iζ )¯ φ ( ϕ φn ( e iζ )) ∗ ! = e i nζ A ne iζ (cid:18) φ (cid:19) , ψ φn ( e iζ ) − ¯ φ ( ψ φn ( e iζ )) ∗ ! = e i nζ A ne iζ (cid:18) − ¯ φ (cid:19) , a direct calculation shows that(5.6) (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) ≤ C j =[ l ( ǫ )]+1 X j =0 (cid:13)(cid:13)(cid:13) A je iζ (cid:13)(cid:13)(cid:13) . We need an explicit upper bound on l ( ǫ ). Sincedet ϕ φn ( e iζ ) ψ φn ( e iζ )¯ φ ( ϕ φn ( e iζ )) ∗ − ¯ φ ( ψ φn ( e iζ )) ∗ ! = − ¯ φϕ φn ( e iζ )( ψ φn ( e iζ )) ∗ − ¯ φψ φn ( e iζ )( ϕ φn ( e iζ )) ∗ anddet ϕ φn ( e iζ ) ψ φn ( e iζ )¯ φ ( ϕ φn ( e iζ )) ∗ − ¯ φ ( ψ φn ( e iζ )) ∗ ! = det( e i nζ A ne iζ (cid:18) φ − ¯ φ (cid:19) ) = ( − φ ) e inζ , and noting that ( ϕ φn ( e iζ )) ∗ = e inζ ϕ φn ( e − iζ ) = e inζ ϕ φn ( e iζ ) and ( ψ φn ( e iζ )) ∗ = e inζ ψ φn ( e iζ ), we have(5.7) ϕ φn ( e iζ ) ψ φn ( e iζ ) + ψ φn ( e iζ ) ϕ φn ( e iζ ) = 2 . For a positive integer L , define f φ = ( ϕ φ , ϕ φ , . . . , ϕ φL , ϕ φL ) , g φ = ( ψ φ , ψ φ , . . . , ψ φL , ψ φL ) . By the Cauchy-Schwarz inequality and (5.7), (cid:13)(cid:13)(cid:13) f φ (cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) g φ (cid:13)(cid:13)(cid:13) L ≥ |h f φ , g φ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X j =1 ϕ φj ψ φj + ϕ φj ψ φj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 L. Therefore, (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) ϕ φ ( e iζ ) (cid:13)(cid:13)(cid:13) L ≥ L, which implies that l ( ǫ ) ≤ √ ǫ − + 1. Thus (5 .
6) turns into (cid:13)(cid:13)(cid:13) ψ φ ( e iζ ) (cid:13)(cid:13)(cid:13) l ( ǫ ) ≤ C cǫ − X (cid:13)(cid:13)(cid:13) A je iζ (cid:13)(cid:13)(cid:13) , and the second inequality in (5.5) becomes | F φ ((1 − ǫ ) e iζ ) | ≤ C sup ≤ s ≤ cǫ − (cid:13)(cid:13) A se iζ (cid:13)(cid:13) , concluding the proof. (cid:3) .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 27 Recall from Subsection 2.2 that for x ∈ T d , µ x and Λ x denote the canon-ical maximal spectral measures of C x , E x , respectively. As a consequence ofProposition 5.2, we have the following: Proposition 5.4.
For any x ∈ T d and any e iζ ∈ ∂ D and < ǫ < , wehave (5.8) µ x ( ζ − ǫ, ζ + ǫ ) ≤ Cǫ sup ≤ s ≤ cǫ − (cid:13)(cid:13) A se iζ (cid:13)(cid:13) and (5.9) Λ x ( ζ − ǫ, ζ + ǫ ) ≤ Cǫ sup s ≤ cǫ − (cid:13)(cid:13) A se iζ (cid:13)(cid:13) , where C > is a universal constant.Proof. As x is fixed throughout the proof, let us drop this parameter fromthe notation. Half-line case:
Let F ( z ) = Z e iθ + ze iθ − z dµ ( θ )be the Carath´eodory function of the (canonical maximal) spectral measure µ of C . Then, ℜ F ( re iζ ) = Z − r r − r cos( θ − ζ ) dµ ( θ ) . For r = 1 − ǫ , we have ℜ F ((1 − ǫ ) e iζ ) = Z ǫ − ǫ − ǫ + ǫ − − ǫ ) cos( θ − ζ ) dµ ( θ ) . Consequently, ℜ F ((1 − ǫ ) e iζ ) ≥ Z ζ + ǫζ − ǫ ǫ − ǫ − ǫ + ǫ − − ǫ )(1 − ǫ ) dµ ( θ ) , which implies that ℜ F ((1 − ǫ ) e iζ ) ≥ ǫ ( µ ( ζ − ǫ, ζ + ǫ )) . By Proposition 5.2 ( φ = 1), the result then follows. Full-line case:
For the full-line matrix E x , if we modify α − into some˜ α − ∈ ∂ D , then E x decouples into two half-line matrices C + x and C − x . Let F + ( z ) , F − ( z ) be the Carath´eodory functions of their spectral measure µ + , µ − respectively. The idea is to apply the CMV version of the Damanik-Killip-Lenz maximum modulus principle argument [15] of Munger-Ong [36], whichuses the Alexandrov family of half-line CMV matrices and the maximummodulus principle to control the Carath´eodory function of E x .Define the anti-Carath´eodory function M − ( z ) = ℜ (1 − ¯ α ) − i ℑ (1 + ¯ α ) F − ( z ) i ℑ (1 − ¯ α ) − ℜ (1 + ¯ α ) F − ( z ) and let G ( z ; k, l ) = h δ k , ( E x − z ) − δ l i . Then,(5.10) | G ( z ; 0 ,
0) + G ( z ; 1 , | ≤ (cid:12)(cid:12)(cid:12)(cid:12) − F + ( z ) M − ( z ) F + ( z ) − M − ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . Moreover, the Carath´eodory function of E x isΦ( z ) = Z e iθ + ze iθ − z d Λ( θ ) , where Λ is the canonical maximal spectral measure (cf. Subsection 2.2).Then,(5.11) Φ( z ) = 1 + 2 z ( G ( z ; 0 ,
0) + G ( z ; 1 , . Write 1 − F + ( z ) M − ( z ) F + ( z ) − M − ( z ) = (1 − M − ( z )+1 M − ( z ) − ) + (1 + M − ( z )+1 M − ( z ) − ) F + ( z )(1 + M − ( z )+1 M − ( z ) − ) + (1 − M − ( z )+1 M − ( z ) − ) F + ( z ) . Since − M − ( z ) is a Carath´eodory function, we have ℜ M − ( z ) <
0, which inturn implies that M − ( z )+1 M − ( z ) − ∈ D . By the maximum modulus principle, (cid:12)(cid:12)(cid:12)(cid:12) − F + ( z ) M − ( z ) F + ( z ) − M − ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup φ ∈ ∂ D (cid:12)(cid:12)(cid:12)(cid:12) (1 − φ ) + (1 + φ ) F + ( z )(1 + φ ) + (1 − φ ) F + ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . Recalling (5.1), we thus have(5.12) (cid:12)(cid:12)(cid:12)(cid:12) − F + ( z ) M − ( z ) F + ( z ) − M − ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup φ ∈ ∂ D (cid:12)(cid:12)(cid:12) F φ + ( z ) (cid:12)(cid:12)(cid:12) , where F φ + is the Carath´eodory function of the Alexandrov measure dµ + ,φ .Therefore, taking z = (1 − ǫ ) e iζ and using (5.11), (5.10), (5.12), and (5.2),we find that | Φ((1 − ǫ ) e iζ ) | ≤ | G ((1 − ǫ ) e iζ ) + G ((1 − ǫ ) e iζ ) |≤ φ ∈ ∂ D | F φ + ((1 − ǫ ) e iζ ) |≤ C sup ≤ s ≤ cǫ − k A se iζ k . A similar argument gives ℜ Φ((1 − ǫ ) e iζ ) ≥ ǫ Λ( ζ − ǫ, ζ + ǫ ) . Together they yield(5.13) Λ( ζ − ǫ, ζ + ǫ ) ≤ Cǫ sup ≤ s ≤ cǫ − k A se iζ k , finishing the proof. (cid:3) .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 29 Absolutely Continuous Spectral Measures.
Since the proof of ab-solute continuity in the half-line case is very similar to (and even easier than)the proof in the full-line case, we only consider the latter scenario and provethe absolute continuity of Λ on Σ.Let us first recall the following consequence of subordinacy theory, com-pare [41, Theorem 10.9.4] and [28, Corollary 2.4].
Theorem 5.5.
Define B = (cid:26) e iζ ∈ ∂ D : lim sup s ≥ k A se iζ k < ∞ (cid:27) , then both µ | B and Λ | B are absolutely continuous. We mention that subordinacy theory was originally developed by Gilbert-Pearson in the setting of half-line continuum Schr¨odinger operators [26]and then extended to the whole-line case by Gilbert [25]. The relevanceof bounded solutions in this context was pointed out by Behnke [7], Simon[39], and Stolz [44].We also need the following discrete analog of a result of Eliasson from[17]:
Theorem 5.6.
Let δ > , ω ∈ DC( κ, τ ) , and A ∈ SL(2 , R ) . Then thereis a constant ǫ = ǫ ( γ, τ, δ, k A k ) such that if A ∈ C ωδ ( T d , SL(2 , R )) is realanalytic with k A − A k δ ≤ ǫ and the rotation number of the cocycle ( ω, A ) satisfies k ρ ( ω, A ) − h n, ω ik R / Z ≥ κ | n | τ ∀ = n ∈ Z d or ρ ( ω, A ) − h n, ω i ∈ Q for some n ∈ Z d , then ( ω, A ) is reducible to con-stant coefficients of a quasi-periodic (perhaps with frequency ω ) and analytictransformation. It is enough to prove that(5.14) Λ(Σ \B ) = 0 . Let R be the set of e iζ ∈ Σ such that the cocycle is reducible. We knowthat
R\B only contains elements e iζ for which ( ω, S ( α, e iζ )) is analyticallyreducible to a constant parabolic cocycle. By Theorem 5.6 it follows that R\B is countable: indeed for any such e iζ , the well known gap labelingtheorem ensures that there exists a k ∈ Z d such that 2 ρ ( ω, S ( α, e iζ )) = h k, ω i mod Z . If e iζ ∈ R , by (2.5), any non-zero solution of E x u = e iζ u satisfies inf n ∈ Z | u n | + | u n +1 | >
0. In particular there are no eigenvalues in R ,thus (using countability of R\B ) Λ(
R\B ) = 0. It therefore suffices to show(5.15) Λ(Σ \R ) = 0 . Let J j ( e iζ ) be an open 2 ǫ j − -neighborhood of e iζ ∈ K j in ∂ D (recall that K j was defined in (3.11)). By Corollary 3.2 and Proposition 5.4 we haveΛ( J j ( e iζ )) ≤ sup ≤ s ≤ Cǫ − j − k A se iζ k | J j ( e iζ ) |≤ sup ≤ s ≤ Cǫ − j − k A se iζ k | J j ( e iζ ) |≤ Cǫ j − , where | · | denotes Lebesgue measure. Take a finite subcover such that K j ⊂ S rl =0 J i ( e iζ l ). Refining this subcover if necessary, we may assume thatevery z ∈ ∂ D is contained in at most 2 different J j ( e iζ l ). By Lemma 4.5 and(2.11), | ρ ( J j ( e iζ )) | = k ( ζ − − ǫ j − , ζ + 2 − ǫ j − ) ≥ cǫ j − . By Lemma 3.3, if e iζ ∈ K j , we have (cid:13)(cid:13)(cid:13) ρ ( e iζ ) − h n j , ω i (cid:13)(cid:13)(cid:13) R / Z ≤ ǫ j − for some | n j | < N j − . This implies that 2 ρ ( K j ) can be covered by 2 N j − open arcs T s of length 2 ǫ j − . Since | T s | ≤ c | ρ ( J j ( e iζ )) | for any s , e iζ ∈ K j ,there are at most 2([ c ] + 1) + 4 open arcs J j ( e iζ l ) such that 2 ρ ( J j ( e iζ l ))intersects T s . We conclude that there are at most 2(2([ c ] + 1) + 4) N j − openarcs J j ( z l ) to cover K j . Then(5.16) Λ( K j ) ≤ r X j =0 Λ( J j ( e iζ l )) ≤ CN j − ǫ j − ≤ Cǫ j − . Since ǫ j = ǫ j and ǫ is small, (5.16) implies(5.17) X j Λ( K j ) < ∞ . Since Σ \R ⊆ lim sup K j , the Borel-Cantelli Lemma and (5.17) imply (5.15).By our earlier discussion, this in turn implies (5.14). Therefore, Λ = Λ | B ,which is purely absolutely continuous.This completes the proof of absolute continuity and, together with ourdiscussion above of discrete eigenvalues in the half-line case, the proof ofTheorem 1.1. Appendix A. A Quantitative Almost Reducibility Result
The following quantitative almost reducibility result from [11, 35] is thebasis of our proof. .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 31
Proposition A.1.
Let ω ∈ DC( κ, τ ) , κ, τ, r > , σ = , S ∈ SU(1 , , f ∈ C ωr ( T d , su(1 , . Then for any r ′ ∈ (0 , r ) , there exist a constant D = D ( κ, τ ) and a numerical constant C such that if (A.1) k f k r ≤ ǫ ≤ D k S k C (cid:18) min (cid:26) , r (cid:27) ( r − r ′ ) (cid:19) C τ , then there exists B ∈ C ωr ′ (2 T d , SU(1 , , S + ∈ SU(1 , , f + ∈ C ωr ′ ( T d , su(1 , such that B ( x + ω )( S e f ( x ) ) B − ( x ) = S + e f + ( x ) . Define N = | ln ǫ | r − r ′ , let spec ( S ) = { e πiρ , e − πiρ } , then we have the followingestimates: • (Non-resonant case): If | ρ − h n, ω i| ≥ ǫ σ holds for any n ∈ Z d with < | n | < N , then k B k r ′ ≤ ǫ , k B k ≤ ǫ , k f + k , k f + k r ′ ≤ e − N ( r − r ′ ) ≤ ǫ , k S + − S k ≤ ǫ • (Resonant case): If | ρ − h n ∗ , ω i| < ǫ σ holds for some n ∗ ∈ Z d with < | n ∗ | < N , then k B k r ′ ≤ C | n ∗ | τ ǫ r ′ r ′− r , k B k ≤ C | n ∗ | τ , k f + k , k f + k r ′ ≤ ǫe − r ′ ǫ − τ . Moreover, deg B = n ∗ and the constant S + can be written as S + = exp (cid:18) it + v + ¯ v + − it + (cid:19) , where t + ∈ R , v + ∈ C with | t + | ≤ ǫ , | v + | ≤ ǫ e − π | n ∗ | r . Appendix B. An Approach to Establishing a Non-PerturbativeResult in the Case d = 1In this appendix, we explain how to establish purely absolutely continuousspectrum for E x in the non-perturbative regime when the underlying torusdimension is given by d = 1. That is, one wishes to strengthen the statementin Theorem 1.1 in such a way that the smallness of λ does not depend on theDiophantine constants of the frequency ω . The approach we explain herewas first developed in [46], we just give a sketch here for completeness.First, we need the following: Theorem B.1. [30, 49]
Let ω ∈ R \ Q , S ∈ SU(1 , , f ∈ C ωr ( T , su(1 , .There exists ˜ ǫ = ˜ ǫ ( S , r ) > , such that if k f k r ≤ ˜ ǫ , then ( ω, S e f ) isalmost reducible. Therefore, let us consider a Szeg˝o cocycle ( ω, S ( e iζ ) e f ( x,ζ ) ) = ( ω, S ( α, e iζ ))that is close to constant. If λ is small, then one can apply Theorem B.1,and there exists Φ ke iζ ∈ C ωr k (2 T , SU(1 , ke iζ ( x + ω ) S ( e iζ ) e f ( x,ζ ) Φ ke iζ ( x ) − = S k ( e iζ ) e f k ( x,ζ ) . If k is large enough, then one can reduce the initial cocycle ( ω, S ( e iζ ) e f ( x,ζ ) )to the perturbative regime defined in Proposition A.1. Moreover, since thespectrum Σ is compact, one can apply the compactness argument from [35,Proposition 5.2] and show that there exists Γ = Γ( α, r ), which is independentof ζ , such that k Φ ke iζ k r k ≤ Γ , (B.1) | deg Φ ke iζ | ≤ C | ln Γ | . (B.2)In other words, after a finite number (that is uniform with respect to e iζ ∈ Σ)of conjugation steps, one can reduce the cocycle to the perturbative regime.Consequently, one can apply Proposition 3.1, and control the growth ofthe cocycles in the resonant set K j . Corollary 3.2 will follow from (B.1),while Lemma 3.3 still follows as a result of (B.2). The rest of the proof willfollow the same way as in the main body of the present paper.We point out that one cannot instead use Eliasson’s perturbative resultfrom [17], since he uses a parameterized KAM method, and there one cannotsuitably control the ζ -dependence of S k ( e iζ ). Appendix C. Quasi-Periodic CMV Matrices With SingularContinuous Spectrum
Let us give a brief discussion of the phenomenon of singular continuousspectrum in the context of analytic quasi-periodic extended CMV matrices.We rely on known results and hence the main purpose is to make explicithow those known results are relevant to this question.Given the well-known parallels between the theory of discrete one-dimen-sional Schr¨odinger operators (and more generally Jacobi matrices) and CMVmatrices, we are naturally guided by what is known in the Schr¨odinger case.Given this perspective, one should single out two mechanisms that produceexamples with singular continuous spectrum in the Schr¨odinger context:(i) the coexistence of positive Lyapunov exponents and Liouville fre-quencies,(ii) the self-duality with respect to Aubry duality. In the C category it is known that generically, the spectrum of an extended quasi-periodic CMV matrix is purely singular continuous: generic absence of point spectrumwas shown in [38] and generic absence of absolutely continuous spectrum was shown in[18]. The corresponding problem for standard analytic quasi-periodic CMV matrices, whileinteresting, is not well understood. The proofs of the results in the extended case do notcarry over to the standard case. .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 33
In approach (i) one needs a result that establishes positive Lyapunovexponents on the spectrum (or an energy region) for all minimal translations.This input ensures purely singular spectrum. In addition one needs to chosethe translation to be of sufficient Liouville nature so that one can excludeeigenvalues via the Gordon lemma [6, 27]. Let us for simplicity considerthe case d = 1. Here, to measure how exponentially Liouvillean ω is, weconsider β ( ω ) := lim sup n →∞ ln q n +1 q n , where p n q n are the continued fraction approximants of ω . This input ensurescontinuous spectrum. As a result one obtains singular continuous spec-trum, and it should be emphasized that due to the presence of positiveLyapunov exponents, the spectral measures are highly singular – they arezero-dimensional and indeed supported by sets of zero capacity [42].In approach (ii) one uses that duality transforms pure point spectruminto absolutely continuous spectrum, and in some weaker sense vice versaas well. Self-duality therefore implies the absence of both. In addition,duality also transforms positive Lyapunov exponents into zero Lyapunovexponents, and hence the singular continuous spectral measures one obtainsin this approach are accompanied by zero Lyapunov exponents due to self-duality [4, 31]. Indeed, they are (expected to be) more regular, with higherdimensionality and (hence the necessity for) larger supports.Turning our attention to the CMV case now, we note that approach (i)can be implemented as there are CMV versions of the positive Lyapunovexponent result [50] and the Gordon lemma [37]. Indeed, let us attempt topush the Gordon lemma aspect to its limit. Based on the sharp Gordonlemma from [6], we have the following: Theorem C.1.
Suppose that α ∈ C ( T , R ) , ω ∈ R \ Q with β ( ω ) > . Thenfor every x ∈ T , E x has purely singular continuous spectrum on the set S = { e iζ ∈ Σ : β ( ω ) > γ ( e iζ ) > } . Proof.
This is a Gordon-type statement that is essentially contained in theproof of [6, Theorem 1.1]. We just give the short argument here for complete-ness as we are dealing with the extended CMV case (whereas [6] considersthe Schr¨odinger case).Given e iζ ∈ S , we have, by a uniform upper semi-continuity and telescop-ing argument (see, e.g., [6, Proposition 3.1]), that for any sufficiently small ǫ >
0, there exists K = K ( ζ, ω, α, ǫ ), which is in particular independent of x , such that for n ≥ K , we havesup x ∈ T k A q n e iζ ( x + q n ω ) − A q n e iζ ( x ) k ≤ e − ( β − γ − ǫ ) q n . (C.1) sup x ∈ T k A − q n e iζ ( x + q n ω ) − A − q n e iζ ( x ) k ≤ e − ( β − γ − ǫ ) q n . (C.2)As a consequence, we can apply the following lemma: Lemma C.2. [5, 6]
Suppose that (C.1) and (C.2) hold. Then we have max (cid:26)(cid:13)(cid:13)(cid:13) A q n e iζ ( x ) (cid:18) ϕ ϕ ∗ (cid:19) (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) A − q n e iζ ( x ) (cid:18) ϕ ϕ ∗ (cid:19) (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) A q n e iζ ( θ ) (cid:18) ϕ ϕ ∗ (cid:19) (cid:13)(cid:13)(cid:13)(cid:27) ≥ √ , where (cid:18) ϕ ϕ ∗ (cid:19) = (cid:18) ϕ ( e iζ ) ϕ ∗ ( e iζ ) (cid:19) is the initial value at e iζ . According to (2.8), (2.1), (2.9), and noting that A ne iζ = e − inζ ˜ S ( α n , e iζ ) · · · ˜ S ( α , e iζ ) , we are able to utilize the transfer matrix A ne iζ to evaluate the norm of (cid:18) s n t n (cid:19) , the solutions of generalized eigenvalue equations E x s = e iζ s . ThusLemma C.2 ensures that for any x ∈ T , S does not contain any eigenval-ues of E x . The result now follows from the Ishii-Pastur Theorem (cf. [41,Theorem 10.5.7]). (cid:3) Remark C.3. (a) Theorem C.1 is the CMV analog of a Schr¨odinger resultthat is known to be optimal: for the almost Mathieu operator, if γ > β ( ω ),then the operator exhibits Anderson localization [6]. One may expect thatTheorem C.1 is optimal as well.(b) We emphasize that the result holds for every phase x ∈ T and thatthe proof exploits the two-sided nature of the problem. In view of the latteraspect, we regard it as an interesting problem to prove a singular continuityresult for standard (one-sided) CMV matrices with analytic quasi-periodicVerblunsky coefficients. (c) To make Theorem C.1 meaningful, one can apply it, for example, tothe specific model studied by Zhang in [50], as he established the positivityof the Lyapunov exponent there and β ( ω ) can be made as large as needed(even infinite) for suitable choices of ω (that will always form a dense G δ set).Approach (ii), establishing purely singular continuous spectrum for self-dual models, has been worked out in the unitary case by Fillman-Ong-Zhang[21], albeit only for almost every phase. It is curious that while the generalduality correspondence between the pp and ac regimes is still absent inthe extended CMV setting (as pointed out repeatedly in this paper), thesomewhat degenerate situation of a self-dual model has been identified in theunitary setting (for quantum walks, which are closely connected to extendedCMV matrices via the CGMV connection [12]). Of course we mean that the desired singular continuity of spectral measures shouldbe established on the essential spectrum of C x . .C. SPECTRUM FOR QUASI-PERIODIC CMV MATRICES 35 References [1] A. Avila, The absolutely continuous spectrum of the almost Mathieuoperator, preprint (arXiv:0810.2965).[2] A. Avila, B. Fayad, R. Krikorian, A KAM scheme for SL(2 , R ) cocycleswith Liouvillean frequencies, Geom. Funct. Anal. (2011), 1001–1019.[3] A. Avila, S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. (2010), 93–131.[4] A. Avila, S. Jitomirskaya, C. Marx, Spectral theory of extendedHarper’s model and a question by Erd¨os and Szekeres, Invent. Math.
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Nonlinearity (2012) 1771–1797. Department of Mathematics, Nanjing University, Nanjing 210093, China
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