Non-perturbative homogeneous spectrum of some quasi-periodic Gevrey Jaocbi operators with weak Diophantine frequencies
aa r X i v : . [ m a t h . D S ] F e b NON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF SOME QUASI-PERIODICGEVREY JAOCBI OPERATORS WITH WEAK DIOPHANTINE FREQUENCIES
KAI TAOA bstract . In this paper, we consider some quasi-periodic Gevrey Jaocbi operators, which can be defined by ananalytic operators with a Gevrey small perturbation potential, with the weak Diophantine frequency. We provethat if the Lyapunov exponent of the original analytic operator is positive and the perturbation is small enough,then the non-perturbative positivity and joint continuity of the Lyapunov exponent, Anderson Localization andhomogeneous spectrum hold for this perturbative Gevrey operators.
1. I ntroduction
In this paper, we study the following quasi-periodic Jacobi operators H ( x , ω ) on l ( Z ):(1.1) ( H ( x , ω ) φ )( n ) = − a ( x + ( n + ω ) φ ( n + − a ( x + n ω ) φ ( n − + V ( x + n ω ) φ ( n ) , n ∈ Z , where V : T → R is a real function called potential, a : T → C is a complex analytic function and notidentically zero, ω is an irrational number called frequency.It is obvious that the characteristic equations H ( x , ω ) φ = E φ can be expressed as(1.2) φ ( n + φ ( n ) ! = a ( x + ( n + ω ) V ( x + n ω ) − E − a ( x + n ω ) a ( x + ( n + ω ) 0 ! φ ( n ) φ ( n − ! . Define(1.3) M ( x , E , ω ) : = a ( x + ω ) V − E − a ( x ) a ( x + ω ) 0 ! and the n-step transfer matrix M n ( x , E , ω ) : = Y k = n M ( x + k ω, E , ω ) . Due to the fact that an analytic function only has finite zeros, M ( x , E , ω ) and M n ( x , E , ω ) make sense almosteverywhere. By the Kingman’s subadditive ergodic theorem, the Lyapunov exponent(1.4) L ( E , ω ) = lim n →∞ L n ( E , ω ) = inf n →∞ L n ( E , ω ) ≥ L n ( E , ω ) = n Z T log k M n ( x , E , ω ) k dx . In [GDSV18], Goldstein-Damanik-Schlag-Voda studied the spectrum of the following discrete quasi-periodic analytic Schr¨odinger operators, which is a special case of (1.1):(1.6) ( S ( x , ω ) φ )( n ) = φ ( n + + φ ( n − + V ( x + n ω ) φ ( n ) , The author was supported by the Fundamental Research Funds for the Central Universities (Grant B200202004) and ChinaPostdoctoral Science Foundation (Grant 2019M650094). where V is a real analytic function. Note that for any irrational ω , the spectrum does not depend on x .So, it is denoted by S ω . They showed that in the supercritical region, which means L ( E , ω ) > S ω ishomogeneous under the assumption that the frequency ω is the strong Diophantine number, i.e. for some α > k n ω k ≥ ι ω n (log | n | + α for all n , . Here, a closed set
S ⊂ R is called homogeneous if there is τ > E ∈ S and 0 < σ ≤ diam( S ),(1.8) |S ∩ ( E − σ, E + σ ) | > τσ. The homogeneity of the spectrum plays an essential role in the inverse spectral theory of almost peri-odic potentials(refer to the fundamental work of Sodin- Yuditskii [SY95, SY97]). It was shown that thehomogeneity of the spectrum implies the almost periodicity of the associated potentials. What’s more, thehomogeneity of the spectrum is deeply related to Deifts conjecture [BDGL15, DGL17], who asked thatwhether the solutions of the KdV equation are almost periodic if the initial data is almost periodic. So afterthe work [GDSV18] was submitted on arXiv in 2015, the research of the homogeneous spectrum becomesa hot spot in our field.In [LYZZ17], Leguil-You-Zhao-Zhou proved that for a measure-theoretically typical analytic potential, S ω is homogeneous with the strong Diophantine frequency. For the following special example AlmostMathieu operators(1.9) ( M ( x , ω ) φ )( n ) = φ ( n + + φ ( n − + λ cos(2 π ( x + n ω )) φ ( n ) , they further proved that if β ( ω ) = λ ,
1, then S ω is homogeneous. Note that for any irrational ω ,there exist its continued fraction approximants n p s q s o ∞ s = , satisfying(1.10) 1 q s ( q s + + q s ) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω − p s q s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < q s q s + . Then, β ( ω ) is defined as the exponential growth exponent of { p s q s } ∞ s = as follows: β ( ω ) : = lim sup s log q s + q s ∈ [0 , ∞ ] . Obviously, β ( ω ) = ω . Lately, Liu-Shi [LS17] extended it to the finite Liouvillefrequency 0 ≤ β ( ω ) < ∞ , but the coupling number λ of the potential V = λ V needs to be very small. Dueto Avila’s global theory of one-frequency cocycles, this operator is in the subcritical regime and has purelyabsolutely continuous spectrum. In the meantime, Shi with Yuan and Jian [JS19, SY19] obtained the similarresults for the extended Harper’s model, which is also a special case of Jacobi operators (1.1) by a ( x ) = λ exp (cid:18) π i (cid:18) x + ω (cid:19)(cid:19) + λ + λ exp (cid:18) − π i (cid:18) x + ω (cid:19)(cid:19) , V ( x ) = cos 2 π x . Recently, Xu-Zhao [XZ20] gave another homogeneity result for the non-critical extended Harper’s with theDiophantine frequency, i.e. for some α > k n ω k ≥ ι ω | n | α for all n , . If we look at the above results carefully, we will find that except the first one [GDSV18] with strongDiophantine frequency (1.7), the others are all about the operators who are in the subcritical regime or whosepotential is the cosine function (In [LYZZ17], they applied the Aubry duality to obtain the homogeneityfrom the cosine potential to the measure-theoretically typical analytic one) with more generic frequency.
ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 3
Moreover, there are many others spectrum results for these operators and frequencies, such as the AndersonLocalization [J99, AJ09], Ten martini problem [P04, AJ09], dry ten martini problem [AJ09, AYZ].As a contrast, there are much less results for the analytic Schr¨dinger operators in the supercritical region,such as Bourgain-Goldstein [BG00] obtained the non-perturbative Anderson Localization with Diophan-tine frequency, and Goldstein-Schlag [GS11] obtained the Canter spectrum with the strong Diophantine.Goldstein-Schlag-Voda [GSV16] attempted to proved the homogeneous spectrum of the multi-frequencySchr¨odinger operators with more generic frequency, but they only succeed on a non full-measured subset of d − dimension Diophantine frequency D d ,ι,α : for some α > d ,(1.12) k n ω k ≥ ι ω | n | α for all n , . The main reason for this situation is that it is very hard to obtain the good enough properties of the finite-volume determinant f n ( x , E , ω ), which is the key in the study of the spectrum, with more generic frequency.Since it is the denominator of the Green function, it need to be not too small. If the potential is the cosine,then it can be handled explicitly via the Lagrange interpolation for the trigonometric polynomial. But forgeneral potential V , it is hard to judge that when it vanishes or not. In [GS08], Goldstein and Schlag use thesubharmonicity, which comes from the analyticity of the potential, and Hilbert transform to obtain the BMOnorm of f n , and then applied the John-Nirenberg inequality to obtain the following called large deviationtheorem with the strong Diophantine ω :(1.13) mes n x ∈ T : (cid:12)(cid:12)(cid:12) log | f n ( x ) | − (cid:10) log | f n | (cid:11)(cid:12)(cid:12)(cid:12) > n δ o ≤ C exp (cid:18) − c δ n (cid:16) δ ( n )0 (cid:17) − (cid:19) . However, a more generic frequency will enormously increase the di ffi culty of the above methods. What’smore, in [ALSZ], Avila-Last-Shamis-Zhou showed that even for the AMO (1.9), its spectrum is not homo-geneous if exp (cid:16) − β ( ω ) (cid:17) < λ < exp (cid:16) β ( ω ) (cid:17) . Therefore, the homogeneous spectrum is a really risky thing,when the frequency is more ”rational”.There is also another observation of above references that the homogeneity is only valid for the analyticpotential. Recently, Cai-Wang [CW20] announced a new breakthrough that for any Diophantine ω ∈ D d ,ι,α ,there exists a k = D α , where D is a numerical constant, such that for any potential V ∈ C k ( T d , R )with k ≥ k , the spectrum S ω is homogeneous if k V k k ≤ ǫ , where ǫ = ǫ ( ι, α, k , d ). Note that underupper assumptions, this operator is always having purely absolutely continuous spectrum, and its Lyapunovexponent is always zero. This is a traditional territory of reducibility in methodology, which can helppeople seek out the edge points of the spectral gaps, where the cocycles are reducible to constant paraboliccocycles. Unfortunately, it can not work in the supercritical regime. To complicate matters further, non-analytic potential needs more elaborate KAM estimations in [CW20] to make the reducibility work, but itdestroys thoroughly the subharmonicity, which is the key in Goldstein-Schlag’s method for the supercriticalregime.So, it is really a challenge that we are trying to obtain the non-perturbative homogeneous spectrum in thisregime for more generic operator, potential and frequency in this paper. Specifically, we will consider thequasi-periodic Jacobi operator (1.1) with the following called weak Diophantine frequency D ι,α :(1.14) k k ω k > ι exp (cid:0) (log | k | ) α (cid:1) , for some α > . What’s more, we choose the potential to be non-analytic. Let V = f + g , where f is a real analytic functionand g ( x ) is a Gevrey function which is considered to be a perturbation. Here we say a C ∞ function v is aGevrey class G s ( T ) for some s >
1, if(1.15) sup x ∈ T | ∂ m v ( x ) | ≤ MK m ( m !) s ∀ m ≥ KAI TAO for some constants M , K ≥
0. In [K05], Klein showed that the condition (1.15) is equivalent to the followingweak exponential decay of the Fourier coe ffi cients of v :(1.16) | ˆ v ( k ) | ≤ M exp (cid:18) − ρ | k | s (cid:19) ∀ k ∈ Z for some constants M , ρ >
0. Thus, we say v ∈ G s , M ,ρ ( T ) when (1.16) holds. Obviously, G s ( T ) = [ ρ> , M > G s , M ,ρ ( T )and if s < s , M < M and ρ > ρ , then G s , M ,ρ ( T ) ⊂ G s , M ,ρ ( T ). Note that by (1.15) or (1.16), theGevrey class G s ( T ) will become to be the class of real analytic functions on T when s =
1. Thus, for any s ≥ G s ( T ) contains all real analytic functions.Now, we begin to state our conclusions. Let L ( E , ω ) be the Lyapunov exponent of the analytic Jacobioperators, which means V = f in (1.1). When V = f + g , we sometimes use the symbols L ( E , ω, g ) and S ω, g , instead of L ( E , ω ) and S ω to emphasize that these quantities depend on the Gevrey perturbation g . Theorem 1.
Let f be a real analytic function on T , ω ∈ D ι,α and L ( E , ω ) ≥ γ > . For any s > ,M > and ρ > , there exists a constant r = r ( γ, f , a , ι, α, s , M , ρ ) such that for any g ∈ G s , M ,ρ ( T ) , ω ∈ D ι,α andE ∈ R , satisfying k g k ∞ + | ω − ω | + | E − E | < r , where k g k ∞ = max x ∈ T | g ( x ) | , it yields that (1.17) L ( E , ω, g ) > γ, and L ( E , ω, g ) is a jointly continuous function of E , ω and g with modulus of continuity (1.18) h ( t ) = (cid:12)(cid:12)(cid:12) log t (cid:12)(cid:12)(cid:12) − h , where h is an absolute large positive constant. For such E , ω and g, if E ∈ S ω, g , then S ω, g has positivemeasure and there exists σ = σ ( γ, ι, α, f , g , a , s , M , ρ ) such that (1.19) |S ω, g ∩ ( E − σ, E + σ ) | ≥ σ/ , for all σ ∈ (0 , σ ] . In particular, (a) If L ( ω , E ) ≥ γ for all E ∈ R , then for any g ∈ G s , M ,ρ ( T ) , ω ∈ D ι,α , satisfying k g k ∞ + | ω − ω | < r ,the spectrum S ω, g is τ -homogeneous with some τ = τ ( γ, ι, α, f , g , a , s , M , ρ, ω ) . (b) If L ( ω , E ) ≥ γ for all E ∈ ( E ′ , E ′′ ) and there exists ǫ > such that for some g ∈ G s , M ,ρ ( T ) , ω ∈ D ι,α , satisfying k g k ∞ + | ω − ω | < r and (1.20) S ω, g ∩ ( E ′ − ǫ, E ′′ + ǫ ) = S ω, g ∩ ( E ′ , E ′′ ) , then S ω, g ∩ ( E ′ , E ′′ ) is either empty or τ -homogeneous with some τ = τ ( γ, ι, α, f , g , a , s , M , ρ, ω, ǫ ) .What’s more, if L ( E , ω ) ≥ γ > for all E ∈ R and ω ∈ D ι,α , then for any fixed x ∈ T and g ∈ G s , M ,ρ ( T ) satisfying k g k ∞ < r , there exists a zero measured set R ( x , g ) such that for any ω ∈ D ι,α \R ( x , g ) , the Jacobioperator H x ,ω, g satisfies Anderson localization, i.e., it has pure point spectrum with exponentially decayingeigenfunctions. It is shown that the non-perturbative AL and homogeneity hold on the neighborhood of the analyticpotential in the Gevrey topology in the supercritical region. Another reason why we choose L ( E , ω ) ≥ γ ,instead of L ( E , ω ) ≥ γ , to be our assumption is that we know when it happens: we obtained in [T18]that if V = λ f , then for any irrational ω , there exists a constant λ = λ ( f , a ) such that for any λ > λ , L ( E , ω ) ≥ log λ for any E . Hence, ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 5
Theorem 2.
Assume that f ( x ) is a real analytic function on T and V = λ f + g. Then there exists λ = λ ( f , a ) such that for any λ > λ , ρ > , M > , s > , α > and ι > , there exists r = r ( λ, f , a , ρ, s , M , ι, α ) suchthat the followings hold for any g ∈ G s , M ,ρ ( T ) satisfying k g k ∞ < r, and ω ∈ D ι,α : (P) The Lyapunov exponent L ( E , ω, g ) is positive for all E ∈ R :L ( E , ω, g ) ≥
910 log λ > . (C) L ( E , ω, g ) is a jointly continuous function of E , ω and g with modulus of continuity (1.18). (AL) For any fixed x ∈ T , there exists a zero measured set R ( x , g ) such that for any ω ∈ D ι,α \R ( x , g ) , theJacobi operator H x ,ω, g satisfies Anderson localization. (H) If E ∈ S ω, g , then there exists σ = σ ( λ, f , a , g , ι, α, s , M , ρ ) such that (1.19) holds for all σ ∈ (0 , σ ] .In particular, S ω, g has positive measure and is τ -homogeneous with some τ = τ ( λ, f , g , a , ι, α, s , M , ρ, ω ) . Now, let’s make an brief introduction of the main methods used in this paper. Generally speaking, wemainly use the technical route in [GDSV18], but our model asks us to make great improvement on theoriginal methods, and create or lead in some other skills. In fact, compared with the Schr¨odinger operator,the Jacobi operator brings us more complicated calculation, but the main di ffi culties we face are mainlycaused by the non-analytic potential and the more generic frequency.After making some preliminaries in Section 2, we construct an induction to get the key lemma, thelarge deviation theorem of our matrix M n ( x , E , ω ) and finite-volume determinant f gn ( x , E , ω ), in Section 3.Specifically, we apply the existing lemmas for the analytic operators and choose the Gevrey perturbative tobe small to obtain the initial step. Then, we use the Avalanche Principle and the polynomial truncation toobtain the inductive step.This (LDT) is valid for the positivity and continuity of the Lyapunov exponent, and the non-perturbativeAL, but not for the homogeneity of the spectrum, since the smallest deviation δ ( n ) is too large for theWegner’s estimate, which estimates the probability that there exists an eigenvalue in some interval ( E − ǫ, E + ǫ ). From (LDT) it follows immediately that if ǫ = exp ( − H ), then this probability is less thanexp (cid:16) − H ( n δ ( n )) − (cid:17) . We find that δ ( n ) is determined mainly by the holomorphic width of the polyno-mial truncation and the frequency, and the latter plays a central role in our model. Generally, it will be largerwhen the irrational frequency is ”more rational”. So this estimate works in [GDSV18] as δ ( n ) = (log n ) α + n for the strong Diophantine number (1.7), fails in [GSV16] as δ ( n ) = n − α + for the Diophantine number(1.11). So, in [GSV16], Goldstein-Schlag-Voda created the covering form of (LDT) to solve this problem.In our model, we need further optimization, since δ ( n ) = exp (cid:16) − (log n ) α (cid:17) for the weak Diophantine number(1.14), which is the largest of these three.Our method comes from an obversion that for any irrational ω , if n ∼ q k where q k is the denominatorof the continued fraction approximants of ω (1.10), then we have the smallest δ ( n ) = log nn . So, in Section4, we first prove the strong Birkho ff ergodic theorem with n ∼ q k . Then, we optimize the inductive stepand obtain the corresponding (LDT). Combining it with the covering form of (LDT), we get the desiredWenger’s estimate in Section 6.On the other hand, our Gevrey potential also brings some other di ffi culties, such as estimating the numberof the intervals of the exception set of (LDT) in Section 3 and answering the question in Section 5 that whatwill happen when f gn ( x , E , ω ) is far from the Lyapunov exponent. We settle them by truncating this Gevreyfunction into trigonometric functions or polynomials, and applying the existing lemmas for the analyticfunctions and the semialgebraic set theory. Of course, there will be some errors, but they are all within ourtolerance. KAI TAO
All the above results are obtained in order to get the stability of spectrum in Section 7. But what we haveat that time is much worse than what in [GDSV18]. So, we need to optimize the original method again,especially the way to produce spectral segments, which are closed to S ω , of considerable size. After that,we will be able to get the homogeneity very directly in this section.At last, we want to talk more about some results associated with the Gevrey potential. In [K05], Kleinstudied the continuity of Lyapunov exponent in E and AL for the Gevrey Schr¨odinger operators with thefollowing called non-degeneracy condition:(1.21) ∀ x ∈ T , ∃ m , s . t . V ( m ) ( x ) , . He obtained the perturbative results for V = λ g and non-perturbative ones for V = g with 1 < s <
2. Webelieve that our method can be valid to get the homogeneous spectrum of his model, since the essentialdi ffi culties have been solved in this current paper. We also believe in the work we are doing that the Cantoror interval spectrum, had be proved for the analytic potential [GS11, GSV19], also hold for the some Gevreyone. 2. P reliminaries In this section, we make more introduction to the Gevrey functions, define some symbols, and presentsome Lemmas and basic tools we will be using.Firstly, for every positive integer n , consider the truncation of the Gevrey function g : g n ( x ) : = X | k |≤ ¯ n ˆ g ( k ) e ikx , where ¯ n = deg g n will be determined later. Note that g n ( x ) is an analytic function on T and g n ( z ) is aholomorphic on the strip T n : = { z : | Im z | ≤ ρ n } , where ρ n = ρ n s − . Indeed, if z = x + iy with | y | < ρ n , then | g n ( z ) | ≤ X | k |≤ ¯ n | ˆ g ( k ) | e | k || y | ≤ M ¯ n X k = exp (cid:18) − ρ | k | s (cid:19) e k | y | < M ¯ n X k = exp (cid:18) − ρ | k | s (cid:19) < C . It is also obvious that for any n ,(2.1) | g ( x ) − g n ( x ) | ≤ M exp (cid:18) − ρ n s (cid:19) . We choose ¯ n = n s to make the error in (2.1) super exponentially small:(2.2) | g ( x ) − g n ( x ) | ≤ M exp (cid:18) − ρ n (cid:19) . Then, the width of holomorphicity of g n becomes: ρ n = ρ n − s − . Define(2.3) M gn ( x , E , ω ) : = n Y j = a ( x + j ω ) M n ( x , E , ω ) = n Y j = V ( x + j ω ) − E − a ( x + j ω ) a ( x + ( j + ω ) 0 ! . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 7
Note that this matrix M gn ( x , E , ω ) is always Gevrey. However, the analyticity plays a very important role inour paper. So, we need to define the following two analytic matrices related to M gn :(2.4) M tn ( x , E , ω ) = Y j = n E − V n ( x + j ω ) − a ( x + j ω ) a ( x + ( j + ω ) 0 ! , and(2.5) M an ( x , E , ω ) = Y j = n E − f ( x + j ω ) − a ( x + j ω ) a ( x + ( j + ω ) 0 ! , where V n = f + g n . Without loss of generality, f and a has the same width of holomorphicity ρ . Then,for fixed E and ω , M tn ( x , E , ω ) and M an ( x , E , ω ) have their complex analytic extensions on the strip T n and T ρ : = { z = x + iy : | y | < ρ } , respectively. For these matrices, we also consider the quantities L gn , L g , L tn , L an and L a , which are defined analogously to L n and L in (1.5) and (1.4) respectively. It is easy to check that D = L gn ( E , ω ) − L n ( E , ω ) = L g ( E , ω ) − L ( E , ω )(2.6) = L an ( E , ω ) − L n ( E , ω ) = L a ( E , ω ) − L ( E , ω ) , where(2.7) D : = Z T log | a ( x ) | dx = Z T log | a ( x ) | dx < ∞ . Let H [ a , b ] ( x , ω ) be the restriction of H ( x , ω ) to the interval [ a , b ] with Dirichlet boundary conditions, φ ( a − = φ ( b + =
0, and we denote the corresponding Dirichlet determinant by f g [ a , b ] ( x , E , ω ) : = det( H [ a , b ] ( x , ω ) − E ). We use E [ a , b ] j ( x ), ψ [ a , b ] j ( x , · ) to denote the eigenpairs of H [ a , b ] ( x ), with ψ [ a , b ] j ( x , · ) being ℓ -normalized. One has f g [ a , b ] ( x , E , ω ) = f gb − a + ( x + ( a − ω, E , ω ) , where f gn : = f g [1 , n ] . Simple computations yield that(2.8) M gn ( x , E , ω ) = f gn ( x , E , ω ) − a ( x ) f gn − ( x + ω, E , ω ) a ( x + n ω ) f gn − ( x , E , ω ) − a ( x ) a ( x + n ω ) f gn − ( x + ω, E , ω ) ! , and similar communication also holds between f tn and M tn , where f tn ( x + ω, E , ω ) : = det (cid:0) H tn ( x , ω ) − E (cid:1) and H tn ( x , ω ) denotes the restriction operators H n ( x , ω ) with the potential V n .In this paper, we always assume that k g k ∞ < E ∈ E , where E : = [ − k a ( x ) k ∞ − k f ( x ) k ∞ − , k a ( x ) k ∞ + k f ( x ) k ∞ + , as the spectrum S ω, g ⊂ E . Thus, for any n ≥
1, irrational ω and E ∈ E ,(2.9) sup E ∈E , z ∈ T ρ u an ( z , E , ω ) , sup E ∈E , z ∈ T n u tn ( z , E , ω ) , sup E ∈E , x ∈ T u gn ( x , E , ω ) ≤ M , where u an ( z , E , ω ) : = n log k M an ( z , E , ω ) k , u tn ( z , E , ω ) : = n log k M tn ( z , E , ω ) k , u gn ( x , E , ω ) = n log k M gn ( x , E , ω ) k and(2.10) M : = log (cid:16) k a k L ∞ ( T ρ ) + k f k L ∞ ( T ρ ) + (cid:17) . KAI TAO
We applied the analyticities of f and a , which implies the subharmonicities of u an and f an , to obtain thefollowing several propositions in [GT20]. Proposition 2.1 (Theorem 1.3 and Lemma 3.1 in [GT20]) . Let ω ∈ D ι,α and L ( E , ω ) ≥ γ > . Thereexist c = c ( f , a , ι, α ) , N = N ( f , a , ι, α ) and absolute constant C such that for any integer n ≥ N and δ > δ ( n ) : = exp (cid:16) − (log n ) α (cid:17) , (2.11) mes ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log k M an ( x , E , ω ) k − L an ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ ) < exp ( − c δ n ) , and (2.12) mes (cid:26) x ∈ T : (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) f an ( x , E , ω ) (cid:12)(cid:12)(cid:12) − D log (cid:12)(cid:12)(cid:12) f an ( · , E , ω ) (cid:12)(cid:12)(cid:12)E(cid:12)(cid:12)(cid:12)(cid:12) > n δ (cid:27) ≤ C exp (cid:16) − c δ ( δ ( n )) − (cid:17) . Moreover, the set on the left-hand side of (2.12) is contained in the union of less than C n intervals, whereC = C ( f , a ) . Lemma 2.1 (Lemma 3.2 and Lemma 3.15 in [GT20]) . Let ω ∈ D ι,α and L ( E , ω ) ≥ γ > . There exists aconstant C = C ( f , a , ι, α, γ ) such that for any n ≥ , (2.13) (cid:12)(cid:12)(cid:12)(cid:12)D log (cid:12)(cid:12)(cid:12) f an ( · , E , ω ) (cid:12)(cid:12)(cid:12)E − nL an ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C , and (2.14) 0 ≤ L n ( E , ω ) − L ( E , ω ) = L an ( E , ω ) − L a ( E , ω ) < C (cid:0) log n (cid:1) n . Remark . So D n log | f n ( · , E , ω ) | E in (2.12) can be replaced by L an ( E , ω ) or L a ( E , ω ). Lemma 2.2 (Lemma 3.12 in [GT20]) . Let ω ∈ D ι,α and L ( E , ω ) ≥ γ > . There exists a constantC = C ( f , a , ι, α, γ ) such that for any n ≥ , (2.15) sup x ∈ T , | y |∈ ρ/ log k M an ( x + iy , E , ω ) k ≤ nL an ( E , ω ) + C n δ ( n ) , and (2.16) sup x ∈ T , | y |∈ ρ/ n X j = log | a ( x + iy + j ω ) | ≤ nD + C n δ ( n ) . Proposition 2.2 (Theorem 1.4 in [GT20]) . Let u : Ω → R be a subharmonic function on a domain Ω ⊂ C and ω ∈ D ι,α . Suppose that ∂ Ω consists of finitely many piece-wise C curves, T ρ ⊂ Ω and sup z ∈ T ρ u ( z ) < S .Then, there exist constants c = c ( ι, α ) and C = C ( ι, α ) such that for any positive n and δ > C S ρ δ ( n ) , (2.17) mes x ∈ T : | n X k = u ( x + k ω ) − n h u i| > δ n ≤ exp (cid:18) − c ρδ nS (cid:19) . To end this section, we make an introduction to the semialgebraic set theory, which will be applied manytimes in our paper. A set
S ⊂ R d is called semialgebraic if it is a finite union of sets defined by a finitenumber of polynomial equalities and inequalities. More precisely, a semialgebraic set S ⊂ R d is given byan expression S = ∪ j ∩ l ∈ L j { P l k jl } , where { P , . . . , P k } is a collection of polynomials of d variables, L j ⊂ { , . . . , k } and k jl ∈ { >, <, = } . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 9
If the degrees of the polynomials are bounded by p , then we say that the degree of S is bounded by k p . See[B05] for more information on semialgebraic sets.In our context, semialgebraic sets can be introduced by approximating the analytic fucntion h with apolynomial ˜ h . More precisely, given N ≥
1, by truncating h ’s Fourier series and the Taylor series of thetrigonometric functions, one can obtain a polynomial ˜ h of degree less than C ( d , ρ )(1 + log k h k ∞ ) N such that(2.18) sup x ∈ T d | h ( x ) − ˜ h ( x ) | ≤ exp( − N ) . In this paper, we need the following lemma related to this theory.
Lemma 2.3 (Theorem 9.3 in [B05]) . Let
S ⊂ [0 , d be semialgebraic of degree p. Then, the number ofconnected components of S does not exceed k d ( O ( p )) d . Lemma 2.4 (Corollary 9.6 in [B05]) . Let
S ⊂ [0 , d be semialgebraic of degree p. Let ǫ > be a smallnumber and mes ( S ) < ǫ d . Then S may be covered by at most p C (cid:16) ǫ (cid:17) d − balls of radius ǫ . Lemma 2.5 (Corollary 9.7 in [B05]) . Let
S ⊂ [0 , be semi-algebraic of degree B and mes S < η . Let ω ∈ D ι,α and N be a large integer, (2.19) log B ≪ log N ≪ log 1 η . Then, for any x ∈ T , (2.20) ♯ { k = , · · · , N | x + k ω ∈ S ( mod } < N − δ for some δ = δ ( ω ) .Remark . In [B05], Bourgain proved this lemma for the Diophantine ω . Actually, his proof is valid forany irrational ω when (2.19) holds. Lemma 2.6 (Lemma 9.9 in [B05]) . Let
S ⊂ [0 , d be semi-algebraic of degree B and mes S < η , log B ≪ log η . We denote ( ω, x ) ∈ [0 , d × [0 , d the product variable. Fix ǫ > η d . Then there is a decomposition S = S S S , S satisfying Pro j ω S < B C ǫ, and S satisfying the transversality property (2.21) mes d ( S \ L ) < B C ǫ − η d for any n-dimensional hyperplane L s.t. max ≤ j ≤ d − | Pro j L ( e j ) | < ǫ . ldts for G evrey J acobi operators In this section, we are using three subsections to apply the induction to obtain the following LDTs for M gn and f gn . And in the last subsection, we also apply the this proposition to obtain the non-perturbative positiveLyapunov exponent (1.17) and joint continuity (1.18) in Theorem 1. Proposition 3.1.
Let ω ∈ D ι,α and L ( E , ω ) ≥ γ > . Then, there exist N = N ( γ, f , a , ι, α, ρ, s , M ) andr = r ( γ, f , a , ι, α, ρ, s , M ) such that for any g ∈ G s , M ,ρ ( T ) satisfying k g k ∞ < r , n > N , it yields that (3.1) L n ( E , ω, g ) ≥ γ, (3.2) meas (cid:16)n x : (cid:12)(cid:12)(cid:12) u gn ( x , E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) > n δ o(cid:17) < exp (cid:18) − c δ ( δ ( n )) − (cid:19) , (3.3) meas (cid:16)n x : (cid:12)(cid:12)(cid:12) log | f gn ( x , E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) > n δ o(cid:17) < exp (cid:18) − c δ ( δ ( n )) − (cid:19) , and (3.4) (cid:12)(cid:12)(cid:12)(cid:12)D log (cid:12)(cid:12)(cid:12) f gn ( · , E , ω ) (cid:12)(cid:12)(cid:12)E − nL gn ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n ( δ ( n )) . Moreover, the set on the left-hand side of (3.3) is contained in the union of less than n C s intervals, whereC = C ( a , f , g ) is a constant. The initial step.
In this subsection, the main idea is that we will use the existing properties for u an and f an to obtain the initial step by making the perturbative function g small enough.Assume L ( E , ω ) ≥ γ >
0. Then, due to Lemma 2.1, there exists a N = N ( f , a , ι, α, γ ) such that for any n > N ,(3.5) 0 ≤ L n ( E , ω ) − L ( E , ω ) < γ. Combined it with (2.6), we obtain that(3.6) D + L ( E , ω ) = L a ( E , ω ) ≤ L an ( E , ω ) < L a ( E , ω ) + γ = D + L ( E , ω ) + γ. Due to (2.11), we have that there exists a set Q n ( x , E ) satisfying mes Q n ( x , E ) < exp (cid:16) − c γ n (cid:17) such thatfor any x < Q n ( x , E ), k M an ( x , E , ω ) k ≥ exp " D + γ ! n . On the other hand, by (A.2), (cid:12)(cid:12)(cid:12) u gn ( x , E , ω ) − u an ( x , E , ω ) (cid:12)(cid:12)(cid:12) ≤ k g k ∞ exp(( n − S )max n(cid:13)(cid:13)(cid:13) M gn ( x , E , ω ) (cid:13)(cid:13)(cid:13) , k M an ( x , E , ω ) k o , provided the right-hand side is less than 1 / . Let ¯ Q = S n j = n Q j and k g k ∞ ≤ r ( γ, f , a , ι, α, ρ, s , M ) : = γ exp ( − n S ) exp " − D + γ ! n ≪ . Then, mes ¯ Q < exp (cid:16) − c γ n (cid:17) and for any x < ¯ Q and n ∈ [ n , n ],(3.7) (cid:12)(cid:12)(cid:12) u gn ( x , E , ω ) − u an ( x , E , ω ) (cid:12)(cid:12)(cid:12) ≤ γ . Hence, combining it with the H ¨older inequality and the fact that they are both L − integrable with the bound Cn , we have that for any n ′ , n ′′ ∈ [ n , n ],(3.8) (cid:12)(cid:12)(cid:12) L gn ′ ( E , ω ) − L an ′ ( E , ω ) (cid:12)(cid:12)(cid:12) < γ , and then (3.6) implies that(3.9) L gn ′ ( E , ω ) > D + − ! γ, (3.10) (cid:12)(cid:12)(cid:12) L gn ′ ( E , ω ) − L gn ′′ ( E , ω ) (cid:12)(cid:12)(cid:12) < γ + γ < γ . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 11
Combined them with (2.11), we have that(3.11) mes ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ′ log k M gn ′ ( x , E , ω ) k − L gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > γ ) < exp (cid:18) − c γ n ′ (cid:19) . Similarly, we also have that(3.12) mes ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ′ log | f gn ′ ( x , E , ω ) | − L gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > γ ) < exp (cid:18) − c γ n ′ (cid:0) δ ( n ′ ) (cid:1) − (cid:19) . Combining it with the H ¨older inequality again, we obtain that(3.13) (cid:12)(cid:12)(cid:12)(cid:12)D log (cid:12)(cid:12)(cid:12) f gn ′ ( · , E , ω ) (cid:12)(cid:12)(cid:12)E − n ′ L gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ n ′ . The inductive step.
In this part, we first list the following lemmas, which will be applied in the proofof the inductive step:
Proposition 3.2 (Proposition 2.2 in [GS01]) . Let A , . . . , A n be a sequence of × –matrices whose deter-minants satisfy (3.14) max ≤ j ≤ n | det A j | ≤ . Suppose that min ≤ j ≤ n k A j k ≥ µ > n and (3.15) max ≤ j < n [log k A j + k + log k A j k − log k A j + A j k ] <
12 log µ. (3.16) Then (3.17) (cid:12)(cid:12)(cid:12)(cid:12) log k A n · . . . · A k + n − X j = log k A j k − n − X j = log k A j + A j k (cid:12)(cid:12)(cid:12)(cid:12) < C n µ with some absolute constant C. Lemma 3.1 (John-Nirenberg inequality) . Let f be a function of bounded mean oscillation on T . Then thereexist the absolute constants C and c such that for any γ > meas { x ∈ T : | f ( x ) − < f > | > γ } ≤ C exp − c γ k f k BMO ! . Lemma 3.2 (Lemma 2.2 in [GS08]) . Let u : Ω → R be a subharmonic function on a domain Ω ⊂ C .Suppose that ∂ Ω consists of finitely many piece-wise C curves. There exists a positive measure µ on Ω suchthat for any Ω ⋐ Ω (i.e., Ω is a compactly contained subregion of Ω ), (3.19) u ( z ) = Z Ω log | z − ζ | d µ ( ζ ) + h ( z ) , where h is harmonic on Ω and µ is unique with this property. Moreover, µ and h satisfy the bounds µ ( Ω ) ≤ C ( Ω , Ω ) (sup Ω u − sup Ω u ) , (3.20) k h − sup Ω u k L ∞ ( Ω ) ≤ C ( Ω , Ω , Ω ) (sup Ω u − sup Ω u )(3.21) for any Ω ⋐ Ω . Lemma 3.3 (Lemma 2.3 in [BGS01]) . Suppose u is subharmonic on T ρ , with µ ( T ρ ) + sup z ∈ T ρ h ( z ) ≤ S where µ ( T ρ ) and h ( z ) comes from Lemma 3.2. Furthermore, assume that u = u + u , where (3.22) k u − < u > k L ∞ ( T ) ≤ ǫ and k u k L ( T ) ≤ ǫ . Then for some constant C ρ depending only on ρ , k u k BMO ( T ) ≤ C ρ ǫ log S ǫ ! + p S ǫ ! . Now, we present the inductive step in the following lemma:
Lemma 3.4.
Assume for any l ′ , l ′′ ∈ [ n i , n i ] , where n i ≥ n , (3.23) L gl ′ ( E , ω ) > D + − i X k = k γ, (3.24) | L gl ′ ( E , ω ) − L gl ′′ ( E , ω ) | ≤ · i γ, (3.25) meas ( x : (cid:12)(cid:12)(cid:12) u gl ′ ( x , E , ω ) − L gl ′ ( E , ω ) (cid:12)(cid:12)(cid:12) > γ )! < exp (cid:18) − exp (cid:18)(cid:0) log l ′ (cid:1) α (cid:19)(cid:19) , (3.26) meas ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ′ log | f gl ′ ( x , E , ω ) | − L gl ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > γ )! < exp (cid:18) − exp (cid:18)(cid:0) log l ′ (cid:1) α (cid:19)(cid:19) . and (3.27) (cid:12)(cid:12)(cid:12)(cid:12)D log (cid:12)(cid:12)(cid:12) f gl ′ ( · , E , ω ) (cid:12)(cid:12)(cid:12)E − l ′ L gl ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cl ′ . Then for any n ′ , n ′′ ∈ (cid:20) exp (cid:16)(cid:0) log n i (cid:1) α (cid:17) , exp (cid:18)(cid:0) log n i (cid:1) α (cid:19)(cid:21) , we have that (3.28) L gn ′ ( E , ω ) ≥ D + − i + X k = k γ, (3.29) | L gn ′ ( E , ω ) − L gn ′′ ( E , ω ) | ≤ · i + log λ, (3.30) meas (cid:16)n x : (cid:12)(cid:12)(cid:12) u gn ′ ( x , E , ω ) − L gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12) > δ o(cid:17) < exp − c δ exp (cid:0) log n ′ (cid:1) α !! , (3.31) meas ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log | f gn ′ ( x , E , ω ) − L gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ )! < exp − c δ exp (cid:0) log n ′ (cid:1) α !! , and (3.32) (cid:12)(cid:12)(cid:12)(cid:12)D log (cid:12)(cid:12)(cid:12) f gn ′ ( · , E , ω ) (cid:12)(cid:12)(cid:12)E − nL gn ′ ( E , ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n exp − (cid:0) log n ′ (cid:1) α ! . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 13
Proof.
We shall fix ω and E so that they can be suppressed from the notations. To apply Proposition 3.2, weneed to define the following unimodular matrix:(3.33) M un ( x ) = | det M gn ( x ) | M gn ( x ) = | Q nj = d ( x + j ω ) | M gn ( x ) , where d ( x ) = a ( x ) a ( x + ω ). Applying Proposition 2.2 for d ( x ), we have that for any n ≥ δ > δ ( n ),(3.34) mes x ∈ T : | n X k = log | d ( x + k ω ) − nD | > δ n ≤ exp ( − ˇ c δ n ) , where ˇ c = ˇ c ( a , ι, α ). Hence, combined it with (3.25), we have that for any n ∈ [ n i , n i ], we have(3.35) mes ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log k M un ( x ) k − L n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > γ + δ ) < exp (cid:18) − exp (cid:18)(cid:0) log n (cid:1) α (cid:19)(cid:19) + exp ( − ˇ c δ n ) . Here we use the fact that L n = Z T n log k M un ( x ) k dx = L gn − D . Due to (3.23), it yields that L n ≥ − i X k = k γ. Let l = n i and n ∈ (cid:20) exp (cid:16)(cid:0) log n i (cid:1) α (cid:17) , exp (cid:18)(cid:0) log n i (cid:1) α (cid:19)(cid:21) . Then, n = l + ( m − l + l ′ with 2 l ≤ l ′ ≤ l , where(3.36) exp (cid:16)(cid:0) log l (cid:1) α (cid:17) ≤ m ≤ exp (cid:18)(cid:0) log l (cid:1) α (cid:19) . Set A uj ( x ) = M ul ( x + ( j − l ω ), j = , . . . , m −
1, and A um ( x ) = M ul ′ . The matrices A gj have similar definitionsfor M gl . We choose δ = γ in (3.35). Then, there exists a set G i satisfying meas ( T \G ) ≤ m + · exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) ≤ exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) such that for any x ∈ G i , k A uj ( x ) k > exp l γ ! , k A um ( x ) k > exp l ′ γ ! , (cid:12)(cid:12)(cid:12) log k A uj ( x ) k + log k A uj + ( x ) k − log k A uj + ( x ) A uj ( x ) k (cid:12)(cid:12)(cid:12) ≤ l γ, and (cid:12)(cid:12)(cid:12) log k A um − ( x ) k + log k A um ( x ) k − log k A um ( x ) A um − k (cid:12)(cid:12)(cid:12) ≤ l γ. Now the hypothesis of Avalanche Principle are satisfied and hence(3.37) log (cid:13)(cid:13)(cid:13) M un i + ( x ) (cid:13)(cid:13)(cid:13) + m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A uj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) − m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A uj + ( x ) A uj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = O l ! up to a set of measure less than exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) . By the definitions of M un and M gn , easy computationsshow that log (cid:13)(cid:13)(cid:13) M un ( x ) (cid:13)(cid:13)(cid:13) + m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A uj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) − m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A uj + ( x ) A uj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A gj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) − m − X j = log (cid:13)(cid:13)(cid:13)(cid:13) A gj + ( x ) A aj ( x ) (cid:13)(cid:13)(cid:13)(cid:13) . Thus, (3.37) also holds for M gn . If we set u g ( x ) = log (cid:13)(cid:13)(cid:13) A gm ( x ) A gm − ( x ) (cid:13)(cid:13)(cid:13) + log (cid:13)(cid:13)(cid:13) A g ( x ) A g ( x ) (cid:13)(cid:13)(cid:13) , then the previous relation can be rewritten aslog (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + m − X j = log (cid:13)(cid:13)(cid:13) M gl ( x + ( j − l ω ) (cid:13)(cid:13)(cid:13) − m − X j = log (cid:13)(cid:13)(cid:13) M g l ( x + ( j − l ω ) (cid:13)(cid:13)(cid:13) − u g ( x ) = O l ! . Similarly,for any 0 ≤ k < l − (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + m − X j = log (cid:13)(cid:13)(cid:13) M gl ( x + k ω + ( j − l ω ) (cid:13)(cid:13)(cid:13) − m − X j = log (cid:13)(cid:13)(cid:13) M g l ( x + k ω + ( j − l ω ) (cid:13)(cid:13)(cid:13) − u k ( x ) = O l ! , where u gk ( x ) = log (cid:13)(cid:13)(cid:13) M gl ′ − k ( x + k ω + ( m − l ω ) · A gm − ( x + k ω ) (cid:13)(cid:13)(cid:13) + log (cid:13)(cid:13)(cid:13) A g ( x + k ω ) · M gl + k ( x ) (cid:13)(cid:13)(cid:13) , which means that we decrease the length of A gm by k and increase the length of A g by k . Adding theseequations and dividing by l yieldslog (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M gl ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M g l ( x + j ω ) (cid:13)(cid:13)(cid:13) − l − X k = l u gk ( x ) = O l ! up to a set of measure less than exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) . What we have done is to obtain the Dirichlet sums, ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M gl ( x + j ω ) (cid:13)(cid:13)(cid:13) and ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M g l ( x + j ω ) (cid:13)(cid:13)(cid:13) . However, Proposition 2.2 can not be appliedto them, since them are not subharmonic. So, we need to change them by the subharmonic truncationfunctions, ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M tl ( x + j ω ) (cid:13)(cid:13)(cid:13) and ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M t l ( x + j ω ) (cid:13)(cid:13)(cid:13) . Note that up to this set, (cid:13)(cid:13)(cid:13) M gl ( x + j ω ) (cid:13)(cid:13)(cid:13) ≥ exp γ + D ! l ! , (cid:13)(cid:13)(cid:13) M g l ( x + j ω ) (cid:13)(cid:13)(cid:13) ≥ exp γ + D ! l ! . So, due to (A.2), (2.2) and the H ¨older inequality , it yields that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M gl ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M tl ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < exp( − l ) ≪ l , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M g l ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M t l ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < exp( − l ) ≪ l , ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 15 and(3.38) (cid:12)(cid:12)(cid:12) L gl − L tl (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) L g l − L t l (cid:12)(cid:12)(cid:12) ≤ exp − l ! . Obviously, the similar relationships between u gk and u tk also hold. Therefore,log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M tl ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M t l ( x + j ω ) (cid:13)(cid:13)(cid:13) − l − X k = l u tk ( x ) = O l ! up to this set. Recall that l log (cid:13)(cid:13)(cid:13) M tl ( x ) (cid:13)(cid:13)(cid:13) is a subharmonic function on T l with the maximum S and ml ∼ n .So, Proposition 2.2 can be applied with the smallest deviation C S ρ l s − δ ( n ) and we obtain that ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M tl ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M t l ( x + j ω ) (cid:13)(cid:13)(cid:13) = ( m − lL tl − ( m − lL t l + O (cid:16) l s n δ ( n ) (cid:17) up to a set of measure less than 2 exp ( − cn δ ( n )). Note that u tk , k = , . . . , l − n = δ = l s n δ ( n ) ≫
1. So, l − X k = l u tk ( x ) − l − X k = l D u tk E = O (cid:16) l s n δ ( n ) (cid:17) up to a set of measure less than l exp ( − cn δ ( n )). Thus, combining these equations, we have that(3.39) log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) + ( m − lL tl − ( m − lL t l − l − X k = l D u tk E = O (cid:16) l s n δ ( n ) (cid:17) up to a set of measure less than exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) + − cn δ ( n )) + l exp ( − cn δ ( n )). Recalling thedefinitions of δ ( n ) and the setting l , we have that for any n > n (3.40) exp (cid:18) − exp (cid:18)(cid:0) log l (cid:1) α (cid:19)(cid:19) + − cn δ ( n )) + l exp ( − cn δ ( n )) ≪ n − s . Integrating (3.39) and using the H ¨older inequality again, yields(3.41) nL gn + ( m − lL tl − ( m − lL t l − l − X k = l D u tk E = O (cid:16) l s n δ ( n ) (cid:17) . Hence, we obtain (3.28) and (3.29) by the range of m (3.36) and (3.38). Combining (3.41) with (3.39), wehave(3.42) (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) − nL gn (cid:12)(cid:12)(cid:12) = O (cid:16) l s n δ ( n ) (cid:17) up to a set of measure less than n − s . Similarly, due to (3.42), (2.2), (A.2) and H ¨older inequality, we canalso obtain(3.43) (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − nL tn (cid:12)(cid:12)(cid:12) = O (cid:16) l s n δ ( n ) (cid:17) on this set. Let B be this exceptional set and definelog (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − nL tn = u + u where u = B and u = T \ B . Obviously, k u − h u ik L ∞ ( T ) = O (cid:16) l s n δ ( n ) (cid:17) and due to the H ¨olderinequality, k u k L ( T ) ≤ n − s . Due to Lemma 3.3, by choosing S = n s to make ρ be uniform, we obtain that (cid:13)(cid:13)(cid:13) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) BMO ( T ) = O (cid:18) n ( δ ( n )) (cid:19) . Thus, Lemma 3.1 implies us that for any n > n and any δ > n x ∈ T : (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − nL tn (cid:12)(cid:12)(cid:12) > n δ o ≤ C exp (cid:18) − c δ ( δ ( n )) − (cid:19) . Obviously, if (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − nL tn (cid:12)(cid:12)(cid:12) < n δ and δ ≥ ( δ ( n )) , then due to (2.2) and (A.2), (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) ≤ n δ. Choosing δ = ( δ ( n )) and due to the H ¨older inequality again, we obtain that(3.45) (cid:12)(cid:12)(cid:12) L gn − L tn (cid:12)(cid:12)(cid:12) ≤ ( δ ( n )) . Thus, for any δ ≥ ( δ ( n )) ,mes n x ∈ T : (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) − nL gn (cid:12)(cid:12)(cid:12) > n δ o ≤ C exp (cid:18) − c δ ( δ ( n )) − (cid:19) . To obtain (3.31), define " f un ( x ) 00 0 = " M un ( x ) " = : M un ( x ) . and M gn analogously. Obviously, (cid:12)(cid:12)(cid:12) f un ( x ) (cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13) M un ( x ) (cid:13)(cid:13)(cid:13) and (cid:12)(cid:12)(cid:12) f gn ( x ) (cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13) M gn ( x ) (cid:13)(cid:13)(cid:13) . Correspondingly, set A uj ( x ) = M ul ( x + ( j − l ω ), j = , . . . , m − A u ( x ) = M ul ( x ) " = " f ul ( x ) 0 ⋆ , and A um ( x ) = " M ul ′ ( x + ( m − l ω ) = " f ul ′ ( x + ( m − l ω ) ⋆ . It is obvious that (3.35) also holds for A uj ( x ), j = , . . . , m −
1. On the other hand, due to the fact thatlog (cid:12)(cid:12)(cid:12) f ul ( x ) (cid:12)(cid:12)(cid:12) ≤ log (cid:13)(cid:13)(cid:13) A u ( x ) (cid:13)(cid:13)(cid:13) ≤ log (cid:13)(cid:13)(cid:13) M ul ( x ) (cid:13)(cid:13)(cid:13) , (3.25), (3.26), (3.33) and (3.34) , we havemes ( x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l log k A u ( x ) k − L l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > γ + δ ) < exp (cid:18) − exp (cid:18)(cid:0) log l ′ (cid:1) α (cid:19)(cid:19) + exp( − ˇ δ n ) , and an analogous estimate for log (cid:13)(cid:13)(cid:13) A um (cid:13)(cid:13)(cid:13) . Then, the rest proof is very similar to the one for (3.30). And inthis process, we can also get that D log | f gn | E + ( m − lL tl − ( m − lL t l − l − X k = l D ˜ u tk E = O (cid:16) l s n δ ( n ) (cid:17) . Combining it with (3.41) and (3.27), we obtain (3.32). (cid:3)
ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 17
The proofs of Proposition 3.1, Non-perturbative joint continuity.
The proof of Proposition 3.1. (3.9), (3.10), (3.11), (3.12) and (3.13) make the assumptions of Lemma 3.4hold for n . Hence, (3.28), (3.29), (3.30), (3.31) and (3.32) hold for any n ′ , n ′′ ∈ (cid:20) exp (cid:16)(cid:0) log n (cid:1) α (cid:17) , exp (cid:18)(cid:0) log n (cid:1) α (cid:19)(cid:21) .Choose n = h exp (cid:16)(cid:0) log n (cid:1) α (cid:17)i +
1. Then, the assumptions of Lemma 3.4 hold for n , and (3.28), (3.29),(3.30), (3.31) and (3.32) hold for any n ′ , n ′′ ∈ (cid:20) exp (cid:16)(cid:0) log n (cid:1) α (cid:17) , exp (cid:18)(cid:0) log n (cid:1) α (cid:19)(cid:21) . From now on, we choose n i + = h exp (cid:16)(cid:0) log n i (cid:1) α (cid:17)i + i ≥
1. Then, due to Lemma 3.4 and the induction, we obtain that (3.28),(3.29), (3.30), (3.31) and (3.32) hold hold for any n ′ , n ′′ ∈ ∞ S i = (cid:20) exp (cid:16)(cid:0) log n i (cid:1) α (cid:17) , exp (cid:18)(cid:0) log n i (cid:1) α (cid:19)(cid:21) . It is ob-vious that exp (cid:18)(cid:0) log n i (cid:1) α (cid:19) > exp (cid:16)(cid:0) log n i + (cid:1) α (cid:17) . Thus, (3.28), (3.29), (3.30), (3.31) and (3.32) hold for any n > n . Choosing N = n , we finish the proof for the inequalities.At last, we apply the semialgebraic set theory to obtain the number of the intervals. Recall that f , a and g n are all 1-periodic nonconstant real analytic functions on R . Then, due to the semialgebraic sets theory,there exist their truncation polynomials ˜ f n , ˜ a n and ˜ g n of degree less than Cn s , satisfyingsup x ∈ T (cid:12)(cid:12)(cid:12) ˜ f n − f (cid:12)(cid:12)(cid:12) , sup x ∈ T | ˜ a n − a | , sup x ∈ T | ˜ g n − g n | ≤ exp (cid:16) − n (cid:17) . Hence, we can define the new truncation matrix˜ M tn ( x , E , ω ) = Y j = n E − λ ˜ f n ( x + j ω ) − ˜ g n ( x + j ω ) − ˜ a n ( x + j ω )˜ a n ( x + ( j + ω ) 0 ! and its deviation set ˜ Q t ( x ) = ( x : (cid:12)(cid:12)(cid:12) log | ˜ f tn ( x , E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) > n δ ) , where ˜ f tn ( x , E , ω ) is the (1 , M tn ( x , E , ω ). Obviously, ˜ f tn ( x , E , ω ) = det (cid:16) ˜ H tn ( x , ω ) − E (cid:17) is apolynomial of degree less than n s , where ˜ H tn is the operator with the truncation functions ˜ f , ˜ a and ˜ g n . Let Q g ( x ) = n x : (cid:12)(cid:12)(cid:12) log | f gn ( x , E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) > n δ o , and Q g ( x ) = n x : (cid:12)(cid:12)(cid:12) log | f gn ( x , E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) > n δ o . Due to (A.3), it yields that Q g ⊂ Q t ⊂ Q g . Therefore, we finish this proof by applying Corollary 2.3 with d = (cid:3) Remark . Combining this proof, (3.44) and (3.45), we can also obtain that(3.46) meas (cid:16)n x : (cid:12)(cid:12)(cid:12) u tn ( x , E , ω ) − L tn ( E , ω ) (cid:12)(cid:12)(cid:12) > δ o(cid:17) < exp (cid:18) − c δ ( δ ( n )) − (cid:19) , and(3.47) | L gn − L tn | < ( δ ( n )) . Obviously, these LDTs also hold on the strip ρ n . They will be applied to estimate the new upper bound of M gn ( x , E , ω ) in Corollary 5.1. Now, we can give the brief proof of non-perturbative positive Lyapunov exponent and joint continuityin Theorem 1, since the method is very standard and can be found in many references, such as [T14] and[T18], which are also for Jacobi operators.
Proof.
Assume L ( E , ω ) = γ >
0. Due to Proposition 3.1, there exist N = N ( γ, f , a , ι, α, ρ, s , M ) and r = r ( γ, f , a , ι, α, ρ, s , M ) such that for any g ∈ G s , M ,ρ ( T ) satisfying k g k ∞ < r , n > N , L n ( E , ω , g ) ≥ γ. By the continuity, there exists a constant r = r ( γ, f , a , ι, α, ρ, s , M ) such that for any n ∈ [ N , N ] and | E − E | + | ω − ω | < r , L n ( E , ω, g ) ≥ γ. Then, applying (3.2) and the Avalanche Principle in the standardmethod, we can obtain that(3.48) L ( E , ω, g ) ≥ γ, and for large n ,(3.49) (cid:12)(cid:12)(cid:12) L g ( E , ω, g ) + L gn ( E , ω, g ) − L gn ( E , ω, g ) (cid:12)(cid:12)(cid:12) ≤ exp − c γ exp
12 (log n ) α !! ≤ n − C Hence, we have that for any E , E ∈ [ E − ǫ, E + ǫ ], ω , ω ∈ D ( ι, α ) T [ ω − ǫ , ω + ǫ ] and g , g ∈G s , M ,ρ ( T ), satisfying that | ω − ω | + | E − E | + k g − g k L ∞ ≤ r , | L ( E , ω , g ) − L ( E , ω , g ) | ≤ (cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) | E − E | + | ω − ω | + k g − g k L ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − C . (cid:3) Remark . In the proof of (3.49), we can also obtain that (cid:12)(cid:12)(cid:12) L gn ′ ( E , ω ) + L gn ( E , ω ) − L gn ( E , ω ) (cid:12)(cid:12)(cid:12) ≤ exp − c γ exp
12 (log n ) α !! , where n ′ ∼ n C . So, for large n ,(3.50) (cid:12)(cid:12)(cid:12) L g ( E , ω, g ) − L gn ( E , ω, g ) (cid:12)(cid:12)(cid:12) < n − ≪ ( δ ) , and(3.51) (cid:12)(cid:12)(cid:12) L gn ( E , ω , g ) − L gn ( E , ω , g ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) | E − E | + | ω − ω | + k g − g k L ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − C . They will also be applied to estimate the new upper bound of M gn ( x , E , ω ) in Corollary 5.1.4. LDT s for n ∼ q k So far, we have obtained the basic lemmas for the operator (1.1), especially the LDTs for the matricesand the determinants. But, as what we have mentioned in the introduction, these LDTs are not good enoughto obtain the homogeneous spectrum. So in this section, we will improve them for n ∼ q k . From now on,we will always fix our Gevrey perturbative g ( x ) and weak Diophantine frequency ω . Thus in the absence ofambiguity, we suppress them from the symbols and the assumptions for ease.To achieve our target, we need to improve the strong Birkho ff ergodic theorem first. Let { x } = x − [ x ].For any positive integer q , complex number ζ = ξ + i η and 0 ≤ ξ <
1, define(4.1) F q ,ζ ( x ) = X ≤ k < q log |{ x + k ω } − ζ | and I ( ζ ) = Z log | y − ζ | dy . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 19
Lemma 4.1.
There exists a constant C, which depends continuously only on ζ ∈ C , such that for any positiveinteger j ≤ , any < σ < and q k , Z exp( σ | F jq k ,ζ ( x ) − jq k I ( ζ ) | ) dx ≤ exp (cid:0) C σ log q k (cid:1) . Proof.
In [T18], our author proved in Lemma 2.2 that for any irrational ω , (cid:12)(cid:12)(cid:12) F q k ,ζ ( x ) − q k I ( ζ ) (cid:12)(cid:12)(cid:12) ≤ C log q k + (cid:12)(cid:12)(cid:12) log |{ x + l ω } − ζ | (cid:12)(cid:12)(cid:12) , where 0 ≤ l < q k such that |{ x + l ω } − ξ | = min q k − l = |{ x + l ω } − ξ | and C = C ( ζ ) is continuous in ζ ∈ C .Thus, (cid:12)(cid:12)(cid:12) F jq k ,ζ ( x ) − jq k I ( ζ ) (cid:12)(cid:12)(cid:12) ≤ C log q k + (cid:12)(cid:12)(cid:12) log |{ x + l ′ ω } − ζ | (cid:12)(cid:12)(cid:12) , where 0 ≤ l ′ < q k such that |{ x + l ′ ω } − ξ | = min q k − l = |{ x + l ω } − ξ | . Due to Lemma 3.2 in [GS01], whichsays that for any finite set Ω ⊂ T and any 0 < σ < Z T exp (cid:0) σ | log dist ( x , Ω ) | (cid:1) dx ≤ σ − σ ( ♯ Ω ) σ , it yields that Z exp( σ | F q k ,ζ ( x ) − q k I ( ζ ) | ) dx ≤ exp (cid:0) σ C log q k (cid:1) · Z T exp (cid:0) σ | log dist ( x − ζ, Ω ) | (cid:1) dx , where Ω = {− l ω : 0 ≤ l < q k } . Obviously, after simple calculations and choosing a new constant C , wefinish this proof. (cid:3) Then we can obtain the following strong Birkho ff Ergodic theorem for n ∼ q k . Proposition 4.1.
Let u : Ω → R be a subharmonic function on a domain Ω ⊂ C . Suppose that ∂ Ω consistsof finitely many piece-wise C curves, T ρ ⊂ Ω and sup z ∈ T ρ u ( z ) < S . Define G ( n ) be a function from N to R ,and (4.2) ˜ G k : = [ j = (cid:8) m ∈ Z : jq k − ( j + G ( q k ) ≤ m ≤ jq k + ( j + G ( q k ) (cid:9) . There exist a constant C = C ( ρ ) such that if G ( q k ) > C log q k , then for any n ∈ ˜ G k and δ ≥ C S G ( q k ) ρ n , (4.3) mes x ∈ T : | n n X l = u ( x + l ω ) − < u > | > δ < exp (cid:18) − ρδ n S (cid:19) . Proof.
Notice that the ergodic measure for the shift on the Torus is the Lebesgue measure and m ( T ) = < u > = R T u ( x ) dx , and n X l = u ( x + l ω ) − n < u > = n X l = Z T ρ log |{ x + l ω } − ζ | d µ ( ζ ) − n Z T ρ I ( ζ ) d µ ( ζ ) + n X l = h ( { x + l ω } ) − n Z h ( y ) dy , where we choose Ω = T ρ here. Choose n = jq k + ¯ G ∈ ˜ G k , where j <
10 and | ¯ G | ≤ ( j + G . Recall that n X l = Z T ρ log |{ x + l ω } − ζ | d µ ( ζ ) = Z T ρ F n ,ζ ( x ) d µ ( ζ ) . Then Z exp σ | n X l = u ( x + l ω ) − n < u > | dx ≤ Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F n ,ζ ( x ) − nI ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx × Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X l = h ( { x + l ω } ) − n Z h ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx . (4.4)Now, we first calculate the part (cid:20)R exp (cid:18) σ (cid:12)(cid:12)(cid:12)(cid:12)R T ρ ( F n ,ζ ( x ) − nI ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dx (cid:21) . Obviously, due to the fact that Z T (cid:12)(cid:12)(cid:12) log | x | (cid:12)(cid:12)(cid:12) dx < ∞ , there exists a constant ˆ C = ˆ C ( ρ ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ (cid:0) log | x − ζ | − I ( ζ ) (cid:1) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( T ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ (cid:0) log | x − ζ | − I ( ζ ) (cid:1) d µ ( ζ ) µ ( T ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ C µ ( T ρ ) . It implies that Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F n ,ζ ( x ) − nI ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F n − jq k ,ζ ( x ) − ( n − jq k ) I ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx × Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F jq k ,ζ ( x ) − jq k I ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ exp (cid:16) σ ˆ C µ ( T ρ )( n − jq k ) (cid:17) Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F jq k ,ζ ( x ) − jq k I ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx . (4.5)On the other hand, since exp( σ · ) is a convex function, the Jensen’s inequality and Lemma 4.1 imply that forany 0 < σ ≤ µ ( Ω ) , Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F jq k ,ζ ( x ) − jq k I ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z Z T ρ exp (cid:16) σµ ( T ρ ) (cid:12)(cid:12)(cid:12) F jq k ,ζ ( x ) − jq k I ( ζ ) (cid:12)(cid:12)(cid:12)(cid:17) d µ ( ζ ) µ ( T ρ ) dx = Z T ρ Z exp (cid:16) σµ ( T ρ ) (cid:12)(cid:12)(cid:12) F jq k ,ζ ( x ) − jq k I ( ζ ) (cid:12)(cid:12)(cid:12)(cid:17) dx d µ ( ζ ) µ ( T ρ ) ≤ Z T ρ exp(2 C ( ζ ) σµ ( T ρ ) log q k ) d µ ( ζ ) µ ( T ρ ) ≤ exp(2 C m σµ ( T ρ ) log q k ) , where C m = sup z ∈ T ρ C ( ζ ), whose existence comes from the compactness of T ρ . Thus, combining it with(4.5) and assuming G > C m
20 ˆ C log q k , we have(4.6) Z exp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ρ ( F n ,ζ ( x ) − nI ( ζ )) d µ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ exp (cid:16) σµ ( T ρ )( ˆ C | ¯ G | + C m log q k ) (cid:17) ≤ exp (cid:16)
40 ˆ C σµ ( T ρ ) G (cid:17) . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 21
Note that log | x | is a subharmonic function. Thus, for the 1-periodic harmonic function h , it must alsohave that(4.7) Z exp( σ | n X k = h ( { x + k ω } ) − n Z h ( y ) dy | ) dx < exp (cid:16)
40 ˆ C σµ ( T ρ ) G (cid:17) . Actually, the estimation of (cid:12)(cid:12)(cid:12)(cid:12)P nk = h ( { x + k ω } ) − n R h ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) is a very historical topic. Our estimation (4.7)can be obtained easily by the well-known property that the k − th Fourier coe ffi cient of the harmonic functionis . exp( −| k | ). (Correspondingly, the one of the subharmonic function is . | k | , which is very bad! This is themain reason that in our field the proofs for the subharmonic functions are very careful and complex , whilethe ones for harmonic functions are easy and can be omitted.) In [AJM17], Avila, Jitomirskaya and Marxobtained a much better conclusion, which can solve a question asked by Erd¨os and Szekeres in 1950. Thereaders interested in this topic can find more details in that paper.Due to (4.4), (4.6) and (4.7), it yields that for any 0 < σ ≤ µ ( T ρ ) , Z exp σ | n X k = u ( x + k ω ) − n < u > | dx < exp (cid:16)
80 ˆ C σµ ( T ρ ) G (cid:17) . Recall the Markov’s inequality: For any measurable extended real-valued function f ( x ) and ǫ > { x ∈ X : | f ( x ) | ≥ ǫ } ) ≤ ǫ Z X | f | dx . Let f ( x ) = exp (cid:16) σ | P nk = u ( x + k ω ) − n < u > | (cid:17) and ǫ = exp( σδ n ), thenmes x ∈ X : | n X k = u ( x + k ω ) − n < u > | > δ n = mes x ∈ X : exp σ | n X k = u ( x + k ω ) − n < u > | ≥ exp( σδ n ) ≤ exp (cid:16) − σδ n +
80 ˆ C σµ ( T ρ ) G (cid:17) . (4.8)Noting that µ ( T ρ ) ≤ S ρ , we finish this proof by choosing σ = ρ S and δ >
100 ˆ
CS Gn ρ . (cid:3) Now, we can obtain the following LDT of f n for n ∼ q k . Lemma 4.2.
Define (4.9) G k : = [ j = (cid:8) m ∈ Z : jq k − G ( q k ) ≤ m ≤ jq k + G ( q k ) (cid:9) where (4.10) G ( q k ) = exp (cid:16)(cid:0) log log q k (cid:1) α (cid:17) . Assume L ( E ) = γ > . Then, for any n ∈ G k , (4.11) mes n x ∈ T : (cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) f gn ( x , E ) (cid:12)(cid:12)(cid:12) − L gn ( E ) (cid:12)(cid:12)(cid:12) > G s ( q k ) H o ≤ C exp ( − c G H ) , where c G = c G ( f , a , ι, α, γ ) . Moreover, the set on the left-hand side is contained in the union of less thann C s intervals. Proof.
This proof is very similar to the one of Lemma 3.4 for f gn ( x ). So, we give the key points in detail andomit the unimportant ones. Let l = [ G ( q k )] be an integer and n = l + ( m − l + l ′ with 2 l ≤ l ′ ≤ l . Then,we can also obtain thatlog | f gn ( x ) | + ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M gl ( x + j ω ) (cid:13)(cid:13)(cid:13) − ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M g l ( x + j ω ) (cid:13)(cid:13)(cid:13) − l − X k = l u gk ( x ) = O l ! up to a set of measure less than 2 n exp (cid:16) − cL l ( δ ( l )) − (cid:17) . Note the fact that n ∈ G ( q k ) implies that ( m − l − , ( m − l − ∈ ˜ G ( q k ) with G ( q k ) ≫ log q k . Similarly, these functions will be replacing by ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M tl ( x + j ω ) (cid:13)(cid:13)(cid:13) and ( m − l − X j = l l log (cid:13)(cid:13)(cid:13) M t l ( x + j ω ) (cid:13)(cid:13)(cid:13) , and then Proposition 4.1 can also be appliedwith δ = Cl s − G ( q k ) ρ n . Similar discussion also holds for P l − k = l u gk ( x ). Hence, we will have that(4.12) log (cid:12)(cid:12)(cid:12) f gn ( x ) (cid:12)(cid:12)(cid:12) + ( m − lL gl − ( m − lL g l − l − X k = l D u gk E = O (cid:16) l s − G ( q k ) (cid:17) up to a set of measure less than 2 n exp (cid:16) − cL l ( δ ( l )) − (cid:17) + l exp ( − cG ( q k )). Recalling the definitions of δ and G ( q k ), the setting l = [ G ( s )] and the fact that n ∼ q k , we have that(4.13) 2 n exp (cid:18) − cL l ( δ ( l )) − (cid:19) + l exp ( − cG ( q k )) ≪ n − s . Thus, (cid:13)(cid:13)(cid:13) log (cid:12)(cid:12)(cid:12) f tn (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) BMO ( T ) = O (cid:16) G s ( q k ) (cid:17) , and due to Lemma 3.1,mes n x ∈ T : (cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13) M tn ( x ) (cid:13)(cid:13)(cid:13) − nL tn (cid:12)(cid:12)(cid:12) > n δ o ≤ C exp (cid:16) − c δ nG − s ( q k ) (cid:17) . At last, we obtain (4.11) by (2.2), and the number of the intervals, similarly. (cid:3) uniform upper estimates and G reen function In this section, we are mainly to build the relationship between f gn ( x , E ) and the distance of E to spec H n ( x ).At first, we apply the subharmonicity to get the uniform upper estimates of M gn and f gn : Corollary 5.1.
Assume L ( E ) > γ . Then, for any | E − E | ≤ H n , where H n = exp (cid:16) ( δ ( n )) − h (cid:17) and h comesfrom (1.18), it yields that (5.1) sup x ∈ T , | y |≤ n − s log | f gn ( x , E ) | ≤ sup x ∈ T , | y |≤ n − s log k M gn ( x , E ) k ≤ nL gn ( E ) + n ( δ ( n )) . Proof.
Define L tn ( y , E ) = n Z T log k M tn ( x + iy , E ) k dx . By Remark 3.1, for any fixed | y | ≤ n − s ,(5.2) mes n x ∈ T : | log k M tn ( x + iy , E ) k − nL tn ( y , E ) | > n δ o < exp (cid:18) − c δ ( δ ( n )) − (cid:19) . Lemma 4.1 in [GS08] proved that for any | y | , | y | ≤ ρ n we have(5.3) (cid:12)(cid:12)(cid:12) L tn ( y , E ) − L tn ( y , E ) (cid:12)(cid:12)(cid:12) ≤ S ρ n | y − y | . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 23
Combining it with (3.51), (3.47) and (5.2), we have that for any | y | < ρ n ( δ ( n )) and | E − E | ≤ exp (cid:16) ( δ ) − h (cid:17) ,(5.4) mes (cid:26) x ∈ T : | log k M tn ( x + iy , E ) k − nL gn ( E ) | > n ( δ ( n )) (cid:27) < exp (cid:18) − c ( δ ( n )) − (cid:19) . Due to the sub-mean value property for subharmonic functions, we have for any | y | < n − s and r > k M tn ( x + iy , E ) k ≤ π r Z D ( x + iy , r ) log k M tn ( z , E ) k dz . Denote by B y ⊂ T the set in (5.4) and choose r = n − s < ρ n ( δ ( n )) . Let B = { z = x + iy ∈ [0 , × ( − r , r ) : x ∈ B y } . Due to (5.4) we have(5.6) 1 π r Z D ( x , r ) \ B log k M tn ( z , E ) k dz ≤ nL gn ( E ) + n ( δ ( n )) . On the other hand, due to the definition of δ ( n ) and the H ¨older inequality, we have that(5.7) 1 π r Z D ( x , r ) T B log k M tn ( z , E ) k d ξ d ζ ≪ n ( δ ( n )) . Combining (5.5), (5.6) and (5.7), we obtain(5.8) sup x ∈ T , | y |≤ n − s log k M tn ( x + iy , E ) k ≤ nL g ( E ) + n ( δ ( n )) . Assume that there exists some x ∈ T , | y | ≤ n − s such thatlog k M gn ( x + iy , E ) k ≥ nL g ( E ) + n ( δ ( n )) . Combined it with (A.2) and (2.2), we get thatlog k M tn ( x + iy , E ) k ≥ nL g ( E ) + n ( δ ( n )) , which is contradictory to (5.8). (cid:3) A consequence of the uniform upper estimate is an improvement of the stability estimations for M gn and f gn . Corollary 5.2.
Assume L ( E ) > γ . Then, for any E , E ∈ [ E − H n , E + H n ] , (5.9) | f gn ( x , E ) − f gn ( x , E ) | ≤ k M gn ( x , E ) − M gn ( x , E ) k ≤ ( | x − x | + | E − E | ) exp (cid:18) nL gn ( E ) + n ( δ ( n )) (cid:19) . In particular, (5.10) | log k M gn ( x , E ) k − log k M gn ( x , E ) k| ≤ ( | x − x | + | E − E | ) exp (cid:16) nL gn ( E ) + n ( δ ( n )) (cid:17) max i | M gn ( x i , E i ) | , (5.11) | log | f gn ( x , E ) | − log | f gn ( x , E ) || ≤ ( | x − x | + | E − E | ) exp (cid:16) nL gn ( E ) + n ( δ ( n )) (cid:17) max i | f gn ( x i , E i ) | , provided the right-hand sides of (5.10) and (5.11) are less than / . Proof.
The proof is completely analogous to that of Lemma A.1. The only di ff erence is that now we can useCorollary 5.1 to bound the M gn and f gn factors. (cid:3) Next we present the key tools to link eigenfunctions of the finite volume operators to (generalized) eigen-functions of a large volume or in infinite volume. They are the
Poisson formula in terms of Green’s functionand a bound on the o ff -diagonal terms of Green’s function in terms of the deviations estimate for the deter-minant f gn ( x , E ). It says that for any solution of the di ff erence equation H ( x ) φ = E φ , we have(5.12) φ ( m ) = ( H [ a , b ] ( x , ω ) − E ) − ( m , a ) φ ( a − + ( H [ a , b ] ( x , ω ) − E ) − ( m , b ) φ ( b + , m ∈ [ a , b ] . Easy computations show that (5.12) also holds if φ satisfies H [ c , d ] ( x ) φ = E φ and [ a , b ] ⊆ [ c , d ]. We denotethis Green’s function by G [ a , b ] ( z , E ) : = (cid:0) H [ a , b ] ( z ) − E (cid:1) − or G n ( z , E ) : = ( H n ( z ) − E ) − . Due to Cramer’s rule,(5.13) G n ( z , E )( k , m ) = f gk − ( z , E ) a ( z + k ω ) ··· a ( z + ( m − ω ) f gn − ( m + ( z + ( m + ω, E ) f gn ( z , E ) , k < m f gk − ( z , E ) f gn − ( m + ( z + ( m + ω, E ) f gn ( z , E ) , k = m f gm − ( z , E ) a ( z + k ω ) ··· a ( z + ( k − ω ) f gn − ( k + ( z + ( k + ω, E ) f gn ( z , E ) , k > m This method was introduced into the theory of localized eigenfunctions in the fundamental work on theAnderson model by Fr¨ohlich and Spencer. Now it will help us address the relationship between the distanceof an energy to the spectrum and the deviation of f gn ( x , E ) to the Lyapunov exponent. Lemma 5.1.
Assume L ( E ) > γ . If (5.14) log (cid:12)(cid:12)(cid:12) f gn ( x , E ) (cid:12)(cid:12)(cid:12) > nL n ( E ) − K , then for any | E − E | < exp (cid:16) − (cid:16) K + n ( δ ( n )) (cid:17)(cid:17) , we have (cid:12)(cid:12)(cid:12) G [1 , n ] ( x , E )( j , k ) (cid:12)(cid:12)(cid:12) ≤ exp (cid:18) − γ | k − j | + K + Cn ( δ ( n )) (cid:19) , (5.15) (cid:13)(cid:13)(cid:13) G [1 , n ] ( x , E ) (cid:13)(cid:13)(cid:13) ≤ exp (cid:18) K + Cn ( δ ( n )) (cid:19) . (5.16) dist( E , spec H n ( x )) ≥ exp (cid:18) − K − Cn ( δ ( n )) (cid:19) . Proof.
Due to (3.50) and (3.51), we have that for any | E − E | ≤ exp (cid:16) − (cid:16) K + n ( δ ( n )) (cid:17)(cid:17) ≪ H n with C large enough, (cid:12)(cid:12)(cid:12) L gn ( E ) − L gn ( E ) (cid:12)(cid:12)(cid:12) ≪ . Combining it with (5.11), we have thatlog | f gn ( x , E ) | ≥ log | f gn ( x , E ) | − | E − E | exp (cid:16) nL g ( E ) + n ( δ ( n )) (cid:17) | f gn ( x , E ) |≥ nL gn ( E ) − K − ≥ nL gn ( E ) − K − . Then, according to (5.13), (2.16) and Corollary 5.1 and the fact that L gn ( E ) − D = L n ( E ) ≥ γ , ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 25 we have for any 1 ≤ k ≤ m ≤ n , |G n ( x , E )( k , m ) | = | f gk − ( x , E ) | · | a ( x + k ω ) · · · a ( x + ( m − ω ) | · | f gn − ( m + ( x + ( m + ω, E ) || f gn ( x , E ) |≤ exp (cid:18) ( k − L gk − ( E ) + k −
1) ( δ ( k − + ( m − k ) D + C ( m − k ) ( δ ( m − k )) ( n − m − L gn − m − ( E ) + n − m −
1) ( δ ( n )) − log | f gn ( x , E ) | (cid:19) ≤ exp (cid:18) − γ | k − m | + K + Cn ( δ ( n )) (cid:19) . Thus, kG n ( x , E ) k ≤ exp (cid:18) K + Cn ( δ ( n )) (cid:19) , which implies that dist( E , spec H n ( x )) = kG n ( x , E ) k − ≥ exp (cid:18) − K − Cn ( δ ( n )) (cid:19) . (cid:3) The above lemma shows that if f gn ( x , E ) is closed to L gn ( E ), then the Green function G n ( x , E ) decay welland has an upper bound, and the distance of E to the spectrum spec H n ( x ) has an lower bound. Naturally,we also want to know what will happen when f gn ( x , E ) is far from L gn ( E ). To answer this question, we needto the Cartan’s estimate from [GS11]. Definition 5.1.
Let H ≥ . For an arbitrary set B ⊂ D ( z , ⊂ C , where D ( z , r ) = { z ∈ C : | z − z | < r } ,we say that B ∈
Car ( H , K ) if B ⊂ j S j = D ( z j , r j ) with j ≤ K, and (5.17) X j r j < e − H . If d ≥ is an integer and B ⊂ d Q j = D ( z j , , ⊂ C d , then we define inductively that B ∈
Car d ( H , K ) if for any ≤ j ≤ d there exists B j ⊂ D ( z j , , ⊂ C , B j ∈ Car ( H , K ) so that B ( j ) z ∈ Car d − ( H , K ) for any z ∈ C \ B j ,here B ( j ) z = n ( z , . . . , z d ) ∈ B : z j = z o . The definition is motivated by the following generalization of the usual Cartan estimate to several vari-ables. Note that given a set S that has a centre of symmetry, we will let α S , α >
0, stand for the set scaledwith respect to its centre of symmetry.
Lemma 5.2 (Lemma 2.4 in [GS11]) . Let ϕ ( z , . . . , z d ) be an analytic function defined on a polydisk P = d Q j = D ( z j , , , z j , ∈ C . Let M ≥ sup z ∈P log | ϕ ( z ) | , m ≤ log | ϕ ( z ) | , z = ( z , , . . . , z d , ) . Given H ≫ there existsa set B ⊂ P , B ∈
Car d (cid:16) H / d , K (cid:17) , K = C d H ( M − m ) , such that (5.18) log | ϕ ( z ) | > M − C d H ( M − m ) for any z ∈ P \ B . Furthermore, when d = we can take K = C ( M − m ) and keep only the disks of B containing a zero of φ in them. Now, we can answer that question as follows:
Lemma 5.3.
Let H ≫ and L ( E ) > γ . If log | f gn ( x , E ) | ≤ nL gn ( E ) − Hn ( δ ( n )) , then there exists a constant C such that dist( E , spec H n ( x )) < C exp − H +
12 ( δ ( n )) − !! . Proof.
Due to (3.3),mes (cid:26) x ∈ T : (cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) f gn ( x , E ) (cid:12)(cid:12)(cid:12) − L gn ( E ) (cid:12)(cid:12)(cid:12) > n ( δ ( n )) (cid:27) ≤ exp (cid:18) − ( δ ( n )) − (cid:19) . Thus, there exists x ′ satisfying | x ′ − x | < exp (cid:16) − ( δ ( n )) − (cid:17) such that(5.19) log | f gn ( x ′ , E ) | > nL gn ( E ) − n ( δ ( n )) . Combining it with (A.3) and (2.2), we have that(5.20) log | f tn ( x ′ , E ) | > nL gn ( E ) − n ( δ ( n )) . Define Ψ ( z ) = f tn x + z exp (cid:16) ( δ ( n )) − (cid:17) | x ′ − x | ( x ′ − x ) , E , which is a complex analytic function on D (0 ,
1) by noting that exp (cid:16) − ( δ ( n )) − (cid:17) ≪ n − s . Let z ′ be such that Ψ ( z ′ ) = f tn ( x ′ , E ). Obviously, | z ′ | ≤ . Due to Corollary 5.1, we have thatsup x ∈ T , | y | < exp (cid:18) − ( δ ( n )) − (cid:19) log k f tn ( x + iy , E ) k ≤ nL gn ( E ) + n ( δ ( n )) . It means that sup z ∈D (0 , log | Ψ ( z ) | < nL gn ( E ) + n ( δ ( n )) . Due to the Cartan’s estimate (5.21), we have that there exists a set
B ⊂ D (0 , B ∈
Car ( H , K ), K = CHn ( δ ( n )) , such that(5.21) log | Ψ ( z ) | > nL gn ( E ) − CHn ( δ ( n )) for any z ∈ D (0 , \ B . It follows that 0 ∈ D ( z j , r j ) ⊂ B with r j < exp( − H ) for some j , and there exists z ′ ∈ D ( z j , r j ) such that Ψ ( z ′ ) =
0. Let z ′′ = x + z ′ exp (cid:18) ( δ ( n )) − (cid:19) | x ′ − x | ( x ′ − x ). Then, E ∈ spec H tn ( z ′′ ) and | z ′′ − x | ≤ exp (cid:16) − (cid:16) H + ( δ ( n )) − (cid:17)(cid:17) . Since H tn is Hermitian, k H tn ( z ) − H tn ( x ) k ≤ C | z − x | , and that if (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) H tn ( x ) − E (cid:17) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) H tn ( z ) − H tn ( x ) (cid:13)(cid:13)(cid:13) < , then H tn ( z ) − E would be invertible. Hence, we havedist( E , spec H tn ( x )) < C exp (cid:18) − (cid:18) H + ( δ ( n )) − (cid:19)(cid:19) . At last, due to the fact that sup x ∈ T d k H tn ( x ) − H n ( x ) k ≤ exp( − n ) , ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 27 we finish this proof. (cid:3)
Its inverse negative proposition will play an important role in our later proof:
Corollary 5.3.
Let L ( E ) > γ > . If (5.22) dist( E , spec( H n ( x )) > C exp ( − H ) , where < H < ( δ ( n )) − , then log (cid:12)(cid:12)(cid:12) f gn ( x , E ) (cid:12)(cid:12)(cid:12) > nL gn ( E ) − Hn ( δ ( n )) .
6. W egner ’ s estimate and spectrum criterion The following elementary observation links the spectra in finite volume to the decay of the Green function.
Lemma 6.1 (Lemma 2.6 in [GDSV18]) . If for any m ∈ [ a , b ] , there exists Λ m = [ a m , b m ] $ [ a , b ] containingm such that (6.1) (1 − h δ a , δ a m i ) (cid:12)(cid:12)(cid:12) G Λ m ( x , E )( a m , m ) (cid:12)(cid:12)(cid:12) + (1 − h δ b , δ b m i ) (cid:12)(cid:12)(cid:12) G Λ m ( x , E )( b m , m ) (cid:12)(cid:12)(cid:12) < , then E < spec H [ a , b ] ( x ) . We refer to the next result as the covering form of LDT.
Lemma 6.2.
Assume L ( E ) > γ > . Suppose for each point m ∈ [1 , n ] there exists an interval I m ⊂ [1 , n ] such that: (1) dist( m , [1 , n ] \ I m ) ≥ | I m | / , (2) | I m | ≥ m ( f , g , a , α, ι, ρ, s , M ) , (3) log | f I m ( x , E ) | > | I m |L | I m | ( E ) − K m , where K m ≤ | I m | / .Then dist (cid:0) E , spec H n ( x ) (cid:1) ≥ exp (cid:18) − max m (cid:26) K m + | I m | ( δ ( | I m | )) (cid:27)(cid:19) . Proof.
We will apply Lemma 6.1 to obtain that for any | E − E | < exp (cid:16) − max m n K m + | I m | ( δ ( | I m | )) o(cid:17) , E < spec H n ( x ). Due to (3) and Lemma 5.1, we have that for any | E − E | < exp (cid:16) − max m n K m + | I m | ( δ ( | I m | )) o(cid:17) , (cid:12)(cid:12)(cid:12) G I m ( x , E )( k , j ) (cid:12)(cid:12)(cid:12) ≤ exp (cid:18) − γ | k − j | + K m + C | I m | ( δ ( | I m | )) (cid:19) . This and assumptions (1) and (2) guarantee that the assumptions of Lemma 6.1 are satisfied, and then wefinish this proof. (cid:3)
Now, combining it with the LDTs for n ∼ q i , Lemma 4.2, we obtain the desired Wegners estimate. Proposition 6.1.
Assume L ( E ) > γ > . Let k , n be integers such that (cid:0) log n (cid:1) + ≤ k = q i ≤ n. Then thereexists N = N ( f , a , ι, α, s , M , ρ ) such that if k ≥ N , then (6.2) mes (cid:8) x ∈ T : dist (cid:0) spec( H n ( x )) , E (cid:1) < exp ( − k ) (cid:9) ≤ exp (cid:18) − k (cid:19) . Moreover, the set on the left-hand side is contained in the union of less than k Cs n intervals. Proof.
Without loss of generality, k is an even number. Then, choose I m in Lemma 6.2 as I m = [ m , k + m ] , m ∈ [1 , k / , [ m − k / , m + k / , m ∈ [ k / + , n − k / − , [ m − k , m ] , m ∈ [ n − k / , n ]and K m = k ( δ ( k )) . Due to (4.11), we have there exists a set B n , E satisfying thatmes ( B N , E ) < n exp (cid:18) − c k ( δ ( k )) G − s ( k ) (cid:19) such that for any x < B N , E , all assumptions of Lemma 6.2 are satisfied. Noting that k ≥ k ( δ ( k )) ≥ k ( δ ( k )) G − s ( k ) ≫ k − , we have (6.2) and k ≥ (cid:0) log n (cid:1) + . And the number of the intervals comes fromLemma 4.2 obviously. (cid:3) An important consequence of Wegners estimate is that the graphs of the eigenvalues cannot be too flat.
Proposition 6.2.
Assume L ( E ) > γ > . If S ∈ T is connected and mes ( S ) ≥ exp (cid:16) − k (cid:17) for some ksatisfying that (cid:0) log n (cid:1) + ≤ k = q i ≤ n, then mes ( E jn ( S )) ≥ exp ( − k ) for any j ∈ { , · · · , n } such that L ( E ) ≥ γ on E jn ( S ) .Proof. By the continuity of the functions E ( n ) j ( x ), E [ − n , n ] j ( S ) is an interval. Let E be the center of thisinterval. Then if mes ( E jn ( S )) < exp ( − k ), which means that for any x ∈ S , (cid:12)(cid:12)(cid:12)(cid:12) E nj ( x ) − E (cid:12)(cid:12)(cid:12)(cid:12) < exp ( − k ) , then it contradict with (6.2). (cid:3) On the other hand, Lemma 6.1 can be also used to establish our criterion for an energy to be in thespectrum. For this we will use the following well-known fact.
Lemma 6.3 (Lemma 2.7 in [GDSV18]) . If there exist δ > and sequences a k → −∞ , b k → ∞ such that dist( E , spec H [ a k , b k ] ( x )) ≥ δ, then dist( E , spec H ( x )) ≥ δ. We can now formulate the following called spectrum criterion lemma:
Lemma 6.4.
Let L ( ω ) ≥ γ > . If for any x ∈ T , there exists r ( x ) ∈ [ − n / , n / such that dist( E , spec H r ( x ) + [ − n , n ] ( x )) ≥ exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) , then dist( E , S ω ) ≥
12 exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 29
Proof.
Fix x ∈ T and let ¯ n ≥ n be arbitrary. Let p = − ¯ n − n + r ( x − ¯ n ω ) , q = ¯ n + n + r ( x + ¯ n ω ) . We will use Lemma 6.1 to show that ˜ E < spec H [ p , q ] ( x ) for any | ˜ E − E | ≤ exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) . Note that H r ( x − ¯ n ω ) + [ − n , n ] ( x − ¯ n ω ) = H − ¯ n + r ( x − ¯ n ω ) + [ − n , n ] ( x ) = H [ p , p + n ] ( x ) . So, from the hypothesis we infer thatdist( E , spec H [ p , p + n ] ( x )) ≥ exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) . Then, dist( ˜ E , spec H [ p , p + n ] ( x )) ≥
12 exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) . Due to Corollary 5.3 and Lemma 5.1, it yields that (cid:12)(cid:12)(cid:12) G [ p , p + n ] ( x , ˜ E )( j , k ) (cid:12)(cid:12)(cid:12) ≤ exp − γ | j − k | + Cn δ ( n ) + C n (cid:0) log n (cid:1) . It implies (cid:12)(cid:12)(cid:12) G [ p , p + n ] ( x , ˜ E )( p + n , m ) (cid:12)(cid:12)(cid:12) < m ∈ [ p , p + n + [ n / (cid:12)(cid:12)(cid:12) G [ q − n , q ] ( x , ˜ E )( q − n , m ) (cid:12)(cid:12)(cid:12) < m ∈ [ q − n − [ n / , q ]. For m ∈ [ p + n + [ n / , q − n − [ n / a m = m − n + r ( x + m ω ) ≥ p + (cid:20) n (cid:21) + r ( x + m ω ) ≥ p , b m = m + n + r ( x + m ω ) ≤ q − (cid:20) n (cid:21) + r ( x + m ω ) ≤ q . By the hypothesis, Lemma 5.3, Lemma 5.1 and the fact that | m − a m | , | m − b m | ≥ n , b m − a m = n , we get (cid:12)(cid:12)(cid:12) G [ a m , b m ] ( x , ˜ E )( a m , m ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) G [ a m , b m ] ( x , ˜ E )( b m , m ) (cid:12)(cid:12)(cid:12) < . We can now apply Lemma 6.1 to get that ˜ E < spec H [ p , q ] ( x , ω ). Since ¯ N was arbitrary, it follows that we canchoose sequences a k → −∞ and b k → ∞ such thatdist( E , spec H [ a k , b k ] ( x , ω )) ≥
12 exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) . Then, the conclusion follows from Lemma 6.3. (cid:3)
7. S tability and H omogeneity of the S pectrum , A nderson L ocalization Lemma 7.1.
Let L ( E ) ≥ γ > and E ∈ S ω . Then for large n, there exist j ∈ [ − n , n ] and a segment I, | I | ≥ c exp (cid:16) − (cid:0) log n (cid:1) (cid:17) , centered at a point x , such that (7.1) | E [ − n , n ] j ( x ) − E | ≤ exp (cid:18) − c exp (cid:18) (log n ) α (cid:19)(cid:19) , and for any x ∈ I there exists ξ , k ξ k = , with support in [ − n + , n − , such that k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . What’s more, mes E [ − n , n ] j ( I ) > exp (cid:0) − exp (cid:0) (2 log log n ) α (cid:1)(cid:1) . Proof.
Due to the assumption that E ∈ S ω and Lemma 6.4, there exists x ′ ∈ T such that(7.2) max | i |≤ n / dist( E , spec H i + [ − n , n ] ( x ′ )) < exp (cid:18) − (cid:0) δ ( n ) log n (cid:1) − (cid:19) ≤ exp (cid:18) − exp (cid:18) (log n ) α (cid:19)(cid:19) . Let m = exp (cid:16) (3 log log n ) α (cid:17) . Then, due to the definition of D ι,α , there must exist m = q i satisfying(7.3) m ≤ m ≤ exp (cid:0) (log m ) α (cid:1) = (cid:0) log n (cid:1) . Choose l = exp (cid:16) m α (cid:17) . Obviously,(7.4) exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19) ≤ l ≤ exp (cid:18) (log n ) α (cid:19) . Proposition 6.1 implies that for n = l and k = m and then we obtain that n x ∈ T : dist( E , spec H l ( x )) < exp (cid:16) − (cid:0) log l (cid:1) α (cid:17)o ⊂ k [ k = I k , where k ≤ m Cs l ≤ l and I k are intervals such that | I k | ≤ exp (cid:16) − (log l ) α (cid:17) . Due to the definition of D ι,α again, we have k l ω k > ι exp (cid:16)(cid:0) log l (cid:1) α (cid:17) ≫ exp (cid:18) − (log l ) α (cid:19) . Thus, each I k contains at most one point of the form x ′ + ( k − n ) ω with 0 ≤ k < l , and the same conclusionholds for x ′ + ( k + n − l + ω . So, we have that there exists | k ′ | ≤ l ≪ n such that x ′ + ( k ′ − n ) ω and x ′ + ( k ′ + n − l + ω are not in any of the I k . It yields that(7.5) dist( E , spec H l ( x ′ + ( k ′ − n ) ω )) , dist( E , spec H l ( x ′ + ( k ′ + n − l + ω )) ≥ exp (cid:16) − (cid:0) log l (cid:1) α (cid:17) . Recalling (7.2), we have thatdist( E , spec H k ′ + [ − n , n ] ( x ′ )) ≤ exp (cid:18) − exp (cid:18) (log n ) α (cid:19)(cid:19) . Thus, there exists j such that (cid:12)(cid:12)(cid:12)(cid:12) E − E [ − n , n ] j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp (cid:18) − exp (cid:18) (log n ) α (cid:19)(cid:19) , where x = x ′ + k ′ ω . From this, (7.5) and (7.4), it follows that(7.6) dist( E [ − n , n ] j ( x ) , spec H l ( x − n ω )) , dist( E [ − n , n ] j ( x ) , spec H l ( x + ( n − l + ω )) ≥
12 exp (cid:16) − (cid:0) log l (cid:1) α (cid:17) . By the fact that(7.7) (cid:12)(cid:12)(cid:12)(cid:12) E ( n ) j ( x ) − E ( n ) j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k H n ( x ) − H n ( x ) k ≤ C ( f , a , g ) | x − x | , (7.6) also holds for any | x − x | ≤ c exp (cid:16) − (cid:0) log l (cid:1) α (cid:17) . Lemma 5.3 implies thatlog (cid:12)(cid:12)(cid:12) f al ( x , E ) (cid:12)(cid:12)(cid:12) > nL a ( E ) − C (log l ) α l exp −
14 (log l ) α ! . By Lemma 5.1, it obtains (cid:12)(cid:12)(cid:12)(cid:12) G l ( x − n ω, E [ − n , n ] j ( x ))( j , k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp − γ | j − k | + l exp −
18 (log l ) α !! . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 31
Similarly, (cid:12)(cid:12)(cid:12)(cid:12) G l ( x + ( n − l + ω, E [ − n , n ] j ( x ))( j , k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp − γ | j − k | + l exp −
18 (log l ) α !! . Due to the Poisson formula, we obtain that(7.8) (cid:12)(cid:12)(cid:12)(cid:12) φ [ − n , n ] j ( x ; k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp( − cl ) , | k | ≥ n − l . Let ¯ I denote the set of x ∈ T satisfying (7.8). Obviously, n x : | x − x | ≤ c exp (cid:16) − (cid:0) log l (cid:1) α (cid:17)o ⊂ ¯ I . Thus, due to (7.4), we can find a segment I ⊂ ¯ I , centered at a point x , satisfying | I | ≥ c exp (cid:16) − (cid:0) log l (cid:1) α (cid:17) ≥ c exp (cid:16) − (cid:0) log n (cid:1) (cid:17) . Let ξ be the normalized projection of φ [ − n , n ] j ( x ) onto the subspace corresponding to the interval [ − n + , n − | φ [ − n , n ] j ( x ; ± n ) | < exp( − cl ), we have that | ξ k | ≤ − cl ) , | k | > n − l . So, (cid:13)(cid:13)(cid:13)(cid:13) ( H ( x ) − E [ − n , n ] j ( x )) ξ (cid:13)(cid:13)(cid:13)(cid:13) = k ξ − n + k + k ξ n − k < − cl ) < exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . At last, there must exist a q i ′ satisfying (cid:0) log n (cid:1) ≤ q i ′ ≤ exp (cid:0) (2 log log n ) α (cid:1) . So, | I | ≥ exp (cid:18) − q i ′ (cid:19) . Then, due to Corollary 6.2 with k = q i ′ ,mes E [ − n , n ] j ( I ) > exp ( − q i ′ ) ≥ exp (cid:0) − exp (cid:0) (2 log log n ) α (cid:1)(cid:1) . (cid:3) Next we address the stability of the spectral segments produced via the previous lemma by induction onscales. The inductive step that will be stated in Lemma 7.3 is essentially similar to Lemma 12.22 of [B05]and Lemma 3.3 of [GDSV18]. Compared to them, we will use the following elementary lemma from basicperturbation theory to shorten ours:
Lemma 7.2 (Lemma 2.40 in [GSV16]) . Let A be a N × N Hermitian matrix. Let E , ǫ ∈ R , ǫ > , andsuppose there exists φ ∈ R N , k φ k = , such that k ( A − E ) φ k < ε. (7.9) Then, there exists a normalized eigenvector ψ of A with an eigenvalue E such thatE ∈ ( E − ε √ , E + ε √ , |h φ, ψ i| ≥ (2 N ) − / . (7.10) Lemma 7.3.
Let L ( E ) ≥ γ > for any E ∈ ( E ′ , E ′′ ) . Let I ⊂ [0 , be an interval and let j ∈ [ − n , n ] . Assumethat E [ − n , n ] j ( I , ω ) ⊂ ( E ′ , E ′′ ) and that for each x ∈ I, there exists ξ , k ξ k = , with support in [ − n + , n − ,such that (7.11) k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Then, there exists n satisfying n ∼ n such that we can partition I into intervals I m , m ≤ n C s , whereC = C ( f , a , g ) , and for each I m , there exists j ∈ [ − n , n ] such that (cid:12)(cid:12)(cid:12)(cid:12) E [ − n , n ] j ( x ) − E [ − n , n ] j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , x ∈ I m , and for each x ∈ I m , there exists ξ , k ξ k = , with support in [ − n + , n − , satisfying k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Proof.
Fix x ∈ I and let ξ be as in (7.11). We have H [ − n , n ] ( x ) ξ = H [ − n , n ] ( x ) ξ, and then k ( H [ − n , n ] ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Due to Lemma 7.2, it yields that there exists j ( x ) ∈ [ − n , n ] such that(7.12) (cid:12)(cid:12)(cid:12)(cid:12) E [ − n , n ] j ( x ) − E [ − n , n ] j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < √ (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , and(7.13) |h ξ, φ [ − n , n ] j ( x ) i| ≥ / p n . Similarly, we can also obtain that there exists l satisfying thatexp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19) ≤ l ≤ exp (cid:18) (log n ) α (cid:19) such that n x ∈ T : dist( E , spec H l ( x )) < exp (cid:16) − (cid:0) log l (cid:1) α (cid:17)o ⊂ k [ k = I k , where k ≤ l and I k are intervals such that | I k | ≤ exp (cid:16) − (log l ) α (cid:17) . Then, there exists 0 < k ′ < l < n suchthat x ′ + ( k ′ − n ) ω and x ′ + ( − k + n − l + ω are not in any of the I k and thereforedist (cid:16) E [ − n , n ] j ( x ) , spec H l ( x + ( k ′ − n ) ω ) (cid:17) , dist (cid:16) E [ − n , n ] j ( x ) , spec H l ( x + ( − k ′ + n − l + ω ) (cid:17) ≥ exp (cid:16) − (log l ) α (cid:17) . Lemma 5.3 implies that (cid:12)(cid:12)(cid:12)(cid:12) G [ k ′ − n , k ′ − n + l ] ( x , E [ − n , n ] j )( j , k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp − γ | j − k | + l exp −
18 (log l ) α !!(cid:12)(cid:12)(cid:12)(cid:12) G [ − k ′ + n − l , − k ′ + n ] ( x , E [ − n , n ] j )( j , k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp − γ | j − k | + l exp −
18 (log l ) α !! . Due to the Poisson formula 5.12, we have that for any m ∈ [ k ′ − n , k ′ − n + l ] ⊂ [ − n , n ], (cid:12)(cid:12)(cid:12) φ [ − n , n ] ( x ; m ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) G [ k ′ − n , k − n + l ] ( x , E [ − n , n ] j ; m , k ′ − n ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) G [ k ′ − n , k ′ − n + l ] ( x , E [ − n , n ] j ; m , k ′ − n + l ) (cid:12)(cid:12)(cid:12)(cid:12) . Thus, if m ∈ [ k ′ − n + l , k ′ − n + l ], then (cid:12)(cid:12)(cid:12) φ [ − n , n ] ( x ; m ) (cid:12)(cid:12)(cid:12) ≤ exp( − cl ) ≤ exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 33
Similarly, we also have it for any m ∈ [ − k ′ + n − l , − k ′ + n − l ]. Thus,(7.14) | φ [ − n , n ] j ( x ; m ) | ≤ exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , n − k ′ − l ≤ | m | ≤ n − k ′ − l . Let k ′′ = n − r − l and η be the normalized projection of φ [ − n , n ] j ( x ) onto the subspace corresponding tothe interval [ − k ′′ , k ′′ ]. Due to the fact that k ′′ ≫ n and (7.13), it implies that k X m = − k (cid:12)(cid:12)(cid:12) φ j ( x , m ) (cid:12)(cid:12)(cid:12) ≥ n X m = − n (cid:12)(cid:12)(cid:12) φ j ( x , m ) (cid:12)(cid:12)(cid:12) ≥ n . Therefore, for any k ′′ − l ≤ | m | ≤ k ′′ ,(7.15) | ξ ( m ) | ≤ √ n exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) ≤ exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , and(7.16) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) H ( x ) − E [ − n , n ] j ( x ) (cid:17) ξ (cid:13)(cid:13)(cid:13)(cid:13) ≤ exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Now we can declare that we have finished this proof. The reason is that the last thing for our proof is toestimate the number of components of the set of phases x that satisfy (7.12) and (7.16). The main idea isto use the semialgebraic set theory to approximate a and the potential f + g by a trigonometric polynomialswith the fixed ω , just as what we have done to obtain the number of the intervals in the proof of Proposition3.1. By recalling the definitions of ˜ f n , ˜ a n , ˜ g n and ˜ H t ( x ), we have that k H ( x ) − ˜ H t ( x ) k ≤ sup x ∈ T n(cid:12)(cid:12)(cid:12) ˜ f n − f (cid:12)(cid:12)(cid:12) , | ˜ a n − a | , | ˜ g n − g | o ≤ exp( − n ) . Then, the method in the proofs of Lemma 4.3 in [GDSV18] and Lemma 12.22 in [B05] can been appliedhere directly. (cid:3)
Now we complete this induction and obtain the stability of the spectral segments.
Lemma 7.4.
Let L ( E ) ≥ γ > for any E ∈ ( E ′ , E ′′ ) . Let I ⊂ [0 , be an interval and let j ∈ [ − n , n ] . Assumethat E [ − n , n ] j ( I , ω ) ⊂ ( E ′ , E ′′ ) and that for each x ∈ I, there exists ξ , k ξ k = , with support in [ − n + , n − ,such that (7.17) k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , where c > is some constant. If c ≤ C ( V , c , a , γ ) ≪ and N ≥ N ( V , ι, A , γ, c ) , then mes ( E [ − n , n ] j ( I ) \ S ω ) < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Proof.
Let n = n . Using Lemma 7.3, we partition I into intervals I m , m ≤ n Cs , and for each I m , thereexists j ∈ [ − n , n ] such that(7.18) (cid:12)(cid:12)(cid:12)(cid:12) E [ − n , n ] j ( x ) − E [ − n , n ] j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − c exp (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , x ∈ I m , and for each x ∈ I m , there exists ξ , k ξ k =
1, with support in [ − n + , n − k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Let E n , = [ m (cid:16) E [ − n , n ] j ( I m ) ⊖ E [ − n , n ] j ( I m ) (cid:17) , where ⊖ denotes the symmetric set di ff erence S ⊖ S : = ( S − S ) [ ( S − S ) . By the continuity of the parametrization of the eigenvalues, m < n Cs , the setting n = n and (7.18), itfollows that mes ( E n , ) ≤ exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Note that (7.17) implies that dist( E , S ω ) < exp (cid:16) − c exp (cid:16) exp (cid:16) (log log n ) α (cid:17)(cid:17)(cid:17) for all E ∈ E [ − n , n ] j ( I ). At thesame time, if E ∈ E [ − n , n ] j ( I ) \ E n , , then E ∈ E [ − n , n ] j ( I m ), for some m , and (7.19) implies thatdist( E , S ω ) < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Let n k = n k . Through iteration we obtain sets E n , k such thatmes ( E n , k ) < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) , and if E ∈ E [ − n , n ] j ( I ) \ S l ≤ k E n , l , thendist( E , S ω ) < exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Therefore, we finish this proof by note that E [ − n , n ] j ( I ) \ S ω ⊂ [ k E n , k . (cid:3) The proof of the homogeneity.
We first prove (1.19). Lemma 7.1 implies there exists a large constant n = n ( f , a , g , α, ι, s , ρ, M ), such that for any n > n there exist j ∈ [ − n , n ] and of a segment I , | I | > exp (cid:16) − c (log n ) (cid:17) ,centered at a point x , such that(7.20) | E [ − N , N ] j ( x ) − E | ≤ exp (cid:18) − c exp (cid:18) (log n ) α (cid:19)(cid:19) , and for any x ∈ I there exists ξ , k ξ k =
1, with support in [ − n + , n − k ( H ( x ) − E [ − n , n ] j ( x )) ξ k < exp (cid:0) − exp (cid:0) (2 log log n ) α (cid:1)(cid:1) . Then, we can apply Lemma 7.4 to getmes ( E [ − N , N ] j ( I ) \ S ω ) ≤ exp (cid:18) − c (cid:18) exp (cid:18) (log log n ) α (cid:19)(cid:19)(cid:19) . Choose σ = exp (cid:16) − c (log n ) (cid:17) . Then, for any σ < σ , there exists some n > n such that exp (cid:16) − c (log n ) (cid:17) ∼ σ , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) E − exp (cid:16) − c (log n ) (cid:17) , E + exp (cid:16) − c (log n ) (cid:17)(cid:17) ∩ S ω (cid:12)(cid:12)(cid:12)(cid:12) ≥ exp (cid:16) − c (log n ) (cid:17) − exp (cid:18) − c exp (cid:18) (log n ) α (cid:19)(cid:19) − exp (cid:0) − exp (cid:0) (2 log log n ) α (cid:1)(cid:1) ≥
12 exp (cid:16) − c (log n ) (cid:17) . ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 35
It is obvious that the homogeneity in (a) comes from (1.19) directly, and τ = ( , σ ≤ σ , σ S ω ) , σ > σ . . For (b), we assume that S ω ∩ ( E ′ , E ′′ ) , ∅ and E ∈ S ω ∩ ( E ′ , E ′′ ). By choosing exp (cid:16) − c (log n ) (cid:17) . ǫ , wehave (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) E − exp (cid:16) − c (log n ) (cid:17) , E + exp (cid:16) − c (log n ) (cid:17)(cid:17) ∩ ( S ω ∩ ( E ′ , E ′′ )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) E − exp (cid:16) − c (log n ) (cid:17) , E + exp (cid:16) − c (log n ) (cid:17)(cid:17) ∩ (cid:0) S ω ∩ (cid:0) E ′ − ǫ, E ′′ + ǫ (cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) E − exp (cid:16) − c (log n ) (cid:17) , E + exp (cid:16) − c (log n ) (cid:17)(cid:17) ∩ S ω (cid:12)(cid:12)(cid:12)(cid:12) ≥
12 exp (cid:16) − c (log n ) (cid:17) . (cid:3) The proof of AL.
We use the standard method from [B05]. This method depends on the semi-algebraictheory and the large deviation theorem. In Section 10 of [B05], Bourgain used it to obtain the AL for theanalytic discrete Schr¨odinger operators with the Diophantine frequency. The key point is that the degrees ofthe frequency and the potential is much less than the reciprocal of the exception measure of the LDT, suchthat Lemma 2.5 and 2.6 work. In our condition, the readers can check that Lemma 2.5 can be applied with B = exp (cid:16) (log n ) α (cid:17) , N = exp (cid:16) (log n ) α (cid:17) and η = exp (cid:16) − exp (cid:16) (log n ) α (cid:17)(cid:17) , and Lemma 2.6 can be applied withthe same B and η , and ǫ = exp (cid:16) − (log n ) α (cid:17) . Then, due to Bourgain’s method, for any fixed x and large n , excluding an ω -set R ( x , n ) of measuremes ( R ( x , n )) ≤ ǫ = exp (cid:16) − (log n ) α (cid:17) , we ensure that each exp (cid:16) (log n ) α (cid:17) ≤ | n | ≤ exp (cid:16) (log n ) α (cid:17) ,(7.21) (cid:12)(cid:12)(cid:12) G [ − n + n , n + n ] ( x , E , ω )( m , m ) (cid:12)(cid:12)(cid:12) ≤ exp (cid:16) − c | m − m | + n − (cid:17) , ∀ m , m ∈ [ − n + n , n + n ] . Define the interval
Λ = [ exp ( (log n ) α ) ≤ n ≤ exp ( (log n ) α )[ − n + n , n + n ] . Application of the resolvent identity, Lemma 10.33 in [B05], permits us then to deduce from (7.21) that |G Λ ( x , E , ω )( m , m ) | ≤ exp (cid:18) − c | m − m | (cid:19) , ∀ | m − m | >
110 exp (cid:16) (log n ) α (cid:17) and hence we obtain the exponential decay of the extended eigenvector φ ( n ) for any exp (cid:16) (log n ) α (cid:17) ≤ n ≤ exp (cid:16) (log n ) α (cid:17) . Therefore, the Anderson Localization of H x ,ω, g holds for any ω ∈ D ι,α \R ( x , g ), where R ( x , g ) = [ N \ n > N R ( x , n )with mes R ( x , g ) = (cid:3) A ppendix A. some distances associated with the multiplications of matrices In this appendix, we study the distances between the functions which are related to the multiplications ofmatrices. It plays a very important role in our paper, since it is the cornerstone that our Gevrey perturbationworks.
Lemma A.1.
Let A , B : T → GL (2 , R ) , and a i j ( x ) and b i j ( x ) be the ( i , j ) elements of Q n − j = A ( x + j ω ) and Q n − j = B ( x + j ω ) respectively. Assume max x ∈ T {k A ( x ) k , k B ( x ) k} ≤ exp( S ) and max x ∈ T k A ( x ) − B ( x ) k < κ . Then,for any x ∈ T , n ∈ Z + and irrational ω , (A.1) (cid:12)(cid:12)(cid:12) a i j ( x ) − b i j ( x ) (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) − n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n κ exp(( n − S ) , (A.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ exp(( n − S )max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13)Q n − j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:27) . and (A.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log (cid:12)(cid:12)(cid:12) a i j ( x ) (cid:12)(cid:12)(cid:12) − n log (cid:12)(cid:12)(cid:12) b i j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ exp(( n − S )max n(cid:12)(cid:12)(cid:12) a i j ( x ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) b i j ( x ) (cid:12)(cid:12)(cid:12)o , provided the right-hand sides of (A.2) and (A.3) are less than / . Proof. (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) − n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) − n − Y j = A ( x + j ω ) B ( x + ( n − ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + · · · + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ( x ) A ( x + ω ) n − Y j = B ( x + j ω ) − A ( x ) n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ( x ) n − Y j = B ( x + j ω ) − n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n κ exp(( n − S ) . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log + (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ON-PERTURBATIVE HOMOGENEOUS SPECTRUM OF GEVREY JAOCBI OPERATORS 37 ≤ κ exp(( n − S ) (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) . Similarly, we can also obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − n log (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − Y j = B ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ exp(( n − S ) (cid:13)(cid:13)(cid:13)(cid:13)Q n − j = A ( x + j ω ) (cid:13)(cid:13)(cid:13)(cid:13) . (cid:3) R eferences [AJ09] A. Avila and S. Jitomirskaya, The Ten Martini Problem,
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Dis. Cont. Dyn. Syst. , (2020), 4777-4800.C ollege of S ciences , H ohai U niversity , 1 X ikang R oad N anjing J iangsu hina Email address ::