Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
aa r X i v : . [ m a t h . D S ] F e b TRANSITION SPACE FOR THE CONTINUITY OF THE LYAPUNOVEXPONENT OF QUASIPERIODIC SCHR ¨ODINGER COCYCLES
LINGRUI GE, YIQIAN WANG, JIANGONG YOU, AND XIN ZHAO
Abstract.
We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schr¨odingercocycles in the Gevrey space G s with s >
2. In contrast, the Lyapunov exponent has been provedto be continuous in the Gevrey space G s with s < G is the transitionspace for the continuity of the Lyapunov exponent. Introduction
Let X be a C r compact manifold, A ( x ) be a SL (2 , R )-valued function on X and ( X, T, µ ) beergodic with µ a normalized T -invariant measure. Dynamical systems on X × R given by( x, w ) → ( T ( x ) , A ( x ) w )are called a SL (2 , R )-cocycle and denoted by ( T, A ). In particular, if X = T n = R n / π Z n and T = T α : x → x + 2 πα with α independent over Q , we call ( T α , A ) a quasiperiodic SL (2 , R )-cocycle,which is simply denoted by ( α, A ). If moreover, A ( x ) = S E,v ( x ) = (cid:18) E − v ( x ) −
11 0 (cid:19) with v ( x ) a 2 π -periodic function in each variable, we call ( α, S E,v ) a quasiperiodic Schr¨odingercocycle.The n -th iteration of the cocycle ( T, A ) is denoted by (
T, A ) n = ( T n , A n ) where A n ( x ) = A ( T n − x ) · · · A ( x ) , n ≥ I , n = 0 A − n ( T n x ) − , n ≤ − . The (maximum) Lyapunov exponent L ( A ) of the cocycle is defined as L ( A ) = lim n →∞ n Z X ln k A n ( x ) k dµ = inf n n Z X ln k A n ( x ) k dµ ≥ . The limit exists and is equal to the infimum since (cid:8)R X ln k A n ( x ) k dµ (cid:9) n ≥ is a subadditive sequence.Moreover, by Kingman’s subadditive ergodic theorem, we also have L ( A ) = lim n →∞ n ln k A n ( x ) k for µ -almost every x ∈ X .Regularity of the Lyapunov exponent (LE) is one of the central subjects in smooth dynamicalsystems, which depends subtly on the base dynamics T and the smoothness of the matrix A . Inthe present paper, we are mainly interested in how the regularity of A affects the continuity ofLE for quasiperiodic SL (2 , R )/Schr¨odinger cocycles. Our motivation comes from the pioneering(opposite) results on the continuity of the Lyapunov exponent in C ω and C ∞ spaces. • For any quasiperiodic SL (2 , R )-cocycle, the Lyapunov exponent is always continuous withrespect to A in C ω topology [2, 15]. • the Lyapunov exponent is not always continuous with respect to A in C ∞ topology [47, 48]. It is well-known that the Gevrey spaces G s , < s < ∞ are between the C ∞ and analytic spaces.Roughly speaking, a 2 π -periodic cocycle map A ∈ G s means that b A ( k ), the Fourier coefficients of A ,decay sub-exponentially like O ( e − k /s h ), while the Fourier coefficients of an analytic function decayexponentially and the Fourier coefficients of a smooth function decay faster than any polynomials.In this paper, we are interested in finding the optimal Gevrey space to ensure the continuity ofthe Lyapunov exponent. More concretely, we prove that the Lyapunov exponent of quasiperiodicSchr¨odinger cocycles is discontinuous in the Gevrey space G s with s >
2. In contrast, the Lyapunovexponent is continuous in G s with s < G is the transition space forthe continuity of the Lyapunov exponent.It is known that a powerful tool to prove the continuity of LE is the large deviation theorem(LDT) and avalanch principle (AP). Our results in some sense show that LDT breaks down forgeneral G s ( S , SL (2 , R ))-cocycles with s >
2. One can also compare our result with the result in [37]where Klein showed LE is continuous with respect to the energies E if the potential is in an openand dense subspace of G s ( S ) with s > C d − δ -small perturbation. In contrast, it was shown that KAMtorus with constant type frequency persists for all C d + δ -small perturbations [43]. Thus C d is thetransition space for the persistence of KAM torus.1.1. Transition phenomena for quasiperiodic Schr¨odinger operators.
The discrete onedimensional quasiperiodic Schr¨odinger operators on ℓ ( Z ) are given by(1.1) ( H λv,x,α u ) n = u n +1 + u n − + λv ( x + nα ) u n , n ∈ Z , where α ∈ R d is the frequency, x ∈ S d is the phase, λ ∈ R is the coupling constant and v ∈ C r ( S d , R )( r = 0 , , · · · , ∞ , ω ) is called the potential. The spectral properties of operator (1.1) is closelyrelated to the Schr¨odinger cocycle ( α, S E,λv ) ∈ S d × C r ( S d , SL (2 , R )). Quasiperiodic Schr¨odingeroperators naturally arise in solid-state physics, describing the influence of an external magneticfield on the electrons of a crystal.Different from random Schr¨odinger operators, an important feature of one dimensional quasiperi-odic operators is that the family { H λv,x,α } λ ∈ R undergoes a so called metal-insulator transition when | λ | changes from small to large. Indeed, besides the metal-insulator transition, various spectraltransition phenomena take place for quasiperiodic operators. Here we give some perfect examples. Example 1.1 (Metal-insulator transition) . Assume α is Diophantine and v ( x ) = 2 cos 2 πx , thefollowing results were given by Jitomirskaya [29] in 1999, • | λ | > H λv,x,α has Anderson localization for a.e. x , • | λ | = 1, H λv,x,α has purely singular continuous for a.e. x , • | λ | < H λv,x,α has purely absolutely continuous spectrum for a.e. x . Example 1.2 (Sharp spectral transition in frequency) . We denote by β ( α ) = lim sup k →∞ − ln k kα k R / Z | k | , α ∈ R is called Diophantine , denoted by α ∈ DC( κ, τ ), if there exist κ > τ > κ, τ ) := (cid:26) α ∈ R : inf j ∈ Z | nα − j | > κ | n | τ , ∀ n ∈ Z \{ } (cid:27) . where k x k R / Z = dist( x, Z ) . Let v ( x ) = 2 cos 2 πx , i.e., the famous almost Mathieu operators, • | λ | > e β ( α ) , H λv,x,α has Anderson localization for a.e. x [4, 33] , • ≤ | λ | < e β ( α ) , H λv,x,α has purely singular continuous for all x [4, 30] , • | λ | < H λv,x,α has purely absolutely continuous spectrum for all x [1]. Example 1.3 (Sharp spectral transition in phase) . We denote by δ ( α, x ) = lim sup k →∞ − ln k x + kα k R / Z | k | . Let v ( x ) = 2 cos 2 πx , i.e., the famous almost Mathieu operators, • | λ | > e δ ( α,x ) , H λv,x,α has Anderson localization for Diophantine α [34], • ≤ | λ | < e δ ( α,x ) , H λv,x,α has purely singular continuous for all α [30, 34]. • | λ | < H λv,x,α has purely absolutely continuous spectrum for all α [1].Although the transition phenomenon is common for quasiperiodic Schr¨odinger operators, how-ever, the exact transition points are usually difficult to obtain as it depends sensitively on thearithmetic properties of the frequency and phase. This paper will give explicit transition space forthe continuity of the Lyapunov exponent. For our purpose, we introduce the following space ofGevrey functions and its topology.For any smooth function f defined on S , let | f | s,K := 4 π k (1 + | k | ) K k ( k !) s | ∂ k f | C ( S ) ,G s,K ( S ) = { f ∈ C ∞ ( S , R ) || f | s,K < ∞} and G s ( S ) = ∪ K> G s,K ( S ). Note that G s,K ( S ) isa Banach space. Obviously, G ( S ) is the space of analytic functions and for any s ≥ G s ( S )is a subspace of the space of smooth functions. We equip G s ( S ) with the usual inductive limittopology. That is, f n converges to f in G s ( S )-topology if and only if | f n − f | s,K → n → ∞ for some K > α ∈ R \ Q is of bounded type if there exists M >
0, such that the continued fractionexpansion of α , denoted by p n /q n satisfying q n +1 ≤ M q n , ∀ n ∈ N . Theorem 1.1.
Assume α is of bounded type, for quasiperiodic Schr¨odinger cocycle ( α, S E,v ) , wehave (1) For any v ∈ G s ( S ) with s < , the Lyapunov exponent is continuous with respect to v in G s -topology. (2) There exists v ∈ G s ( S ) with s > , such that the Lyapunov exponent is discontinuous at v in G s -topology. Remark 1.1.
Part (1) of Theorem 1.1 was recently proved in [17] (Theorem 6.3 of [17]), we listhere for completeness. The main result of the present paper is part (2).
Remark 1.2.
The bounded type α is dense in R .Part (2) of Theorem 1.1 can be obtained in the same way as in [47] (See page 2367, proof ofTheorem 2 in [47]) from the following examples in SL (2 , R )-cocycles. Theorem 1.2.
Consider quasiperiodic SL (2 , R ) -cocycles over S with α being a fixed irrationalnumber of bounded-type. For any s > , there exists a cocycle D s ∈ G s ( S , SL (2 , R )) such that theLyapunov exponent is discontinuous at D s in G s -topology. [4] proved the measure version and [33] proved the arithmetic version, actually [33] proved Anderson localizationholds for Diophantine phases. [4] proved the case β >
0, [30] proved the case α irrational and | λ | = 1. LINGRUI GE, YIQIAN WANG, JIANGONG YOU, AND XIN ZHAO
A Brief review on the continuity of the Lyapunov exponent.
As we mentioned above,both the base dynamics T and the smoothness of the matrix A affect the regularity of the Lyapunovexponent. This has been the object of considerable recent interests, see Viana [45], Wilkinson [51]and the references therein.If the base dynamics has some hyperbolicity, then the Lyapunov exponent is continuous. Forexample, Furstenberg-Kifer [23] and Hennion [28] proved the continuity of the largest LE of i.i.drandom matrices under a condition of almost irreducibility. Bocker and Viana [10] proved continuityof Lyapunov exponents with respect to the cocycle and the invariant probability for random prod-ucts of SL (2 , R ) matrices in the Bernoulli setting. In higher dimensions, continuous dependencewith respect to A of all Lyapunov exponents for i.i.d. random products of matrices in GL ( d, R )was proved by Avila et al. [3]. If the base dynamics is a subshift of finite type or, more generally,a hyperbolic set, then Backes-Brown-Butler [5] proved that the Lyapunov exponents are alwayscontinuous among H¨older continuous fiber-bunched SL (2 , R )-cocycles.If A ∈ C r ( X, SL (2 , R )), it is known that L ( A ) is upper semicontinuous; thus, it is continuous atgeneric A . Especially, it is continuous at A with L ( A ) = 0 and at uniformly hyperbolic cocycles.The most interesting issue is the continuity of L ( A ) at the nonuniformly hyperbolic cocycles, whichis found to depend on the class of cocycles under consideration including its topology. LE wasproved to be discontinuous at any nonuniformly hyperbolic cocycles in C -topology by Furman [22](Continuity at uniformly hyperbolic cocycles is well-known). Motivated by Mane [40, 41], Bochi[9] further proved that any nonuniformly hyperbolic SL (2 , R )-cocycle over a fixed ergodic systemon a compact space, can be arbitrarily approximated by cocycles with zero LE in the C -topology.In this paper, we are interested in the quasiperiodic cocycles. The base system is a rotation onthe torus in this case, things are very complicated: it will depend on the smoothness of A in a verysensitive way. If the cocycle is analytic, the H¨older continuity of the Lyapunov exponent in thepositive Lyapunov exponent regime was proved by Goldstein and Schlag [25] assuming that α isstrong Diophantine. Similar results were proved in [14] by Bourgain, Goldstein, and Schlag whenthe underlying dynamics is a shift or skew-shift of a higher-dimensional torus. For more results ofthis favor, here is a partial list [12, 19–21, 24–27, 39, 42, 44, 49, 52, 53]. Later, it was proved byBourgain-Jitomirskaya in [15] that the LE is joint continuous for SL (2 , R ) cocycles, in frequency andcocycle map, at any irrational frequencies. Jitomirskaya-Koslover-Schulteis [31] got the continuityof LE with respect to potentials for a class of analytic quasiperiodic M (2 , C )-cocycles. Bourgain [13]extended the results in [15] to multi-frequency case. Jitomirskaya-Marx [35] extended the resultsin [15] to all (including singular) M (2 , C )-cocycles. More recently, continuity of the Lyapunovexponents for one-frequency analytic M ( m, C ) cocycles was given by Avila-Jitomirskaya-Sadel [2].Weak H¨older continuity of the Lyapunov exponents for multi-frequency GL ( m, C )-cocycles, m ≥ , was recently obtained by Schlag [44] and Duarte-Klein [20]. For the lower regularity case, Klein [37]proved that for Schr¨odinger operators with potentials in a Gevrey class G s with 1 ≤ s <
2, the LE isweak H¨older continuous on any compact interval of the energy provided that the frequency is strongDiophantine and the LE is large than 0. While if we further lower the regularity of the potential,Wang-You [47] constructed examples to show that the LE of quasiperiodic Schr¨odinger cocyclescan be discontinuous with respect to the potential even in the C ∞ -topology. Jitomirskaya-Marx[35] obtained similar results in the complex category M (2 , C ) by the tools of harmonic analysis.Recently, Wang-You [48] improved the result in [47] by showing that in C r -topology, 1 ≤ r ≤ + ∞ , there exists Schr¨odinger cocycles with a positive LE that can be approximated by ones with zeroLE. For other results about results on discontinuity of LE, one can see [18, 46].1.3. Outline of the proof and the structure of this paper.
The main results of this paperare based on several improvements of the results in [47] where the authors constructed examples ofdiscontinuity of LE in C ∞ -topology. We first give a quick review of the main ideas. We construct D s ∈ G s ( S , SL (2 , R )) ( s >
2) as the limit of a sequence of cocycles { A n , n = N, N + 1 , ... } in G s ( S , SL (2 , R )) ( s > { A n , n = N, N + 1 , ... } possesses some kind of finitehyperbolic property, that is, k A r + n n ( x ) k ≈ λ r + n for most x ∈ S and λ ≫ r + n → ∞ as n → ∞ ,which gives a lower bound estimate (1 − ε ) ln λ of the Lyapunov exponent of the limit cocycle( α, D s ). Then we modify { A n , n = N, N + 1 , ... } , and construct another sequence of cocycles { e A n , n = N, N + 1 , ... } with some kind of degenerate property such that e A n → D s in G s -topologyas n → ∞ . Moreover, for each n , the Lyapunov exponent of ( α, e A n ) is less than (1 − δ ) ln λ with δ ≫ ε , which implies the discontinuity of the Lyapunov exponent at D s .Compared to [47], the main technical improvements of the present paper are the following twoaspects:(1) Since we need to construct examples in Gevrey space, we need explicit examples of Gevreyfunctions. We find the C ∞ -bump functions are all Gevrey functions based on an opti-mal estimate of the upper bound of its derivatives. Surprisingly, this easy but importantobservation makes it possible for us to construct a counterexample in Gevrey space.(2) Another technical difficulty (the most difficult part) is to prove the sequences { A n } ∞ n = N and { e A n } ∞ n = N converge in G s -topology ( s > C ∞ -topology since one needs very delicate control of the derivatives,and it is out of reach by the methods in [47]. We overcome this difficulty by developing a G s version of the concatenation of finitely many hyperbolic matrices, i.e., Lemma 4.1 andLemma 4.2 in our paper. Our new Lemmas enable us to not only greatly simplify the proofsin [47], but also optimize almost all estimates in [47]. Our construction is optimal since [17]has shown that for the case of s <
2, the Lyapunov exponent is continuous.A key technique in the construction of A n ( x ) comes from Young [54], which was derived fromBenedicks and Carleson [6]. Based on this technique, Wang-Zhang [49] developed a new iterationscheme to prove LDT for Schr¨odinger cocycles with a class of finitely differential potential. Theyproved that for C cos-like (Morse) potential with a large coupling, LE is weak-H¨older continuous.In this aspect, we also give an improvement of the non-resonance lemma proved in [49], which playsan important role in our proof. For more applications of Benedicks-Carleson-Young’s method toquasiperiodic Schr¨odinger operators, we refer readers to [7, 8, 50].The structure of this paper is as follows. In Section 2, we give some basic concepts and prepara-tions for SL (2 , R ) matrices and Gevrey functions. The main idea of the proof will be sketched inSection 3. In Section 4, we give the details of the construction, which is the key part of this paper.Finally, We give the proof of some basic properties of Gevrey functions in Section 5.2. Preparations and some technical lemmas
For θ ∈ S , let R θ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) ∈ SO (2 , R ) . Define the map s : SL (2 , R ) → RP = R / ( π Z )so that s ( A ) is the most contraction direction of A ∈ SL (2 , R ) . That is, for a unit vector ˆ s ( A ) ∈ s ( A ),it holds that k A · ˆ s ( A ) k = k A k − . Abusing the notation a little, let u : SL (2 , R ) → RP = R / ( π Z )be determined by u ( A ) = s ( A − ) and ˆ u ( A ) ∈ u ( A ). Then for A ∈ SL (2 , R ), it is clear that A = R u · (cid:18) k A k k A k − (cid:19) · R π − s , where s, u ∈ [0 , π ) are angles corresponding to the directions s ( A ) , u ( A ) ∈ R / ( π Z ) . LINGRUI GE, YIQIAN WANG, JIANGONG YOU, AND XIN ZHAO
Hyperbolic sequences of SL (2 , R ) -matrices. For a sequence of matrices {· · · A − , A , A , · · · } ,we denote A n = A n − · · · A A and A − n = A − − n · · · A − − . Definition 2.1.
For any 1 < µ ≤ λ , we say the block of matrices { A , A , · · · , A n − } is µ -hyperbolicif(1) k A i k ≤ λ, ∀ i ,(2) k A i k ≥ µ i (1 − ε ) , ∀ i ,and (1) and (2) hold if A , · · · , A n − are replaced by A − n − , · · · , A − .The following lemma is due to Young [54] which tells us when the concatenation of two hyperbolicblocks is still a hyperbolic block. Lemma 2.1 (Lemma 5 of [54]) . Suppose that C satisfies k C k ≥ µ m with µ ≫ . Assume that { A , A , · · · , A n − } is a µ -hyperbolic sequence, and assume that ∠ ( s ( C − ) , s ( A n )) = 2 θ ≪ . Then k A n · C k ≥ µ ( m + n )(1 − ε ) · θ . The Gevrey functions.
Basic properties.
In the following, s ≥ , K > f ∈ G s,K ( I ), we denote | f | s,K = | f | G s,K ( I ) := 4 π k (1 + | k | ) K k ( k !) s | ∂ k f | C ( I ) . Proposition 2.1.
Assume f, g ∈ G s,K ( I ) and ε > is sufficiently small, we have (1) | f g | s,K ≤ | f | s,K | g | s,K . (2) For any ε > , | ∂f | s, (1+ ε s ) K ≤ Kε . (3) Assume | f − | s,K ≤ ε , then (cid:12)(cid:12)(cid:12)(cid:12) f − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ ε s +8 ) K ≤ ε . (4) Assume | f − | s,K ≤ ε , then (cid:12)(cid:12)(cid:12)p f − (cid:12)(cid:12)(cid:12) s, (1+ ε s +16 ) K ≤ ε . (5) Assume | f | s,K ≤ ε , then arcsin( f ) ∈ G s, K ( I ) and | sin f | s, (1+ ε s +8 ) K , | cos f − | s, (1+ ε s +8 ) K ≤ ε . Explicit examples.
Given a C ∞ -bump function, an interesting question is to investigate thedecay rate of its Fourier coefficients (equivalently, the growth rate of its derivatives). In this part, weinvestigate the Gevrey exponent of various C ∞ -bump functions, the proofs will be also postponedto Section 5. We remark that our estimates of the upper bound of the derivatives of C ∞ bumpfunctions are even optimal, we refer readers to [36] for more details. Lemma 2.2.
Assume that < ν < ∞ and f ( x ) = ( e − | x | ν x = 00 x = 0 , then there is some C > such that | f ( n ) ( x ) | ≤ ( C n e − | x | ν ( n !) ν x = 00 x = 0 , ∀ n ∈ N . As a corollary, f ∈ G ν ( R ) . Corollary 2.1.
Assume that < ν < ∞ and f ( x ) = e | x | ν , x = 0 . Then there is some
C > such that for any x = 0 , we have | f ( n ) ( x ) | ≤ C n e | x | ν ( n !) ν , ∀ n ∈ N . Corollary 2.2.
We define a π -periodic function as follows g ( x ) = ( ce − (cid:16) x − c − kπ ) ν + c k +1) π − x ) ν (cid:17) x ∈ ( c + kπ, c + ( k + 1) π )0 x ∈ { c + kπ, c + ( k + 1) π } , then there is some C > such that for any x ∈ S , we have | g ( n ) ( x ) | ≤ C n e − (cid:16) | x − c | ν + | x − c − π | ν (cid:17) ( n !) ν , ∀ n ∈ N . As a corollary, g ∈ G ν ( S ) . Let I n, = h c − q βn , c + q βn i , I n, = h c − q βn , c + q βn i and I n = I n, S I n, . Lemma 2.3.
Assume < ν < , β > and δ > satisfy < β ν − δ < . For any n ≥ N , thereexist an absolute constant C and a π -periodic function f n ∈ G ν ,C ( S ) such that f n ( x ) = 1 x ∈ I n ∈ (0 , x ∈ I n \ I n
10= 0 x ∈ S \ I n and | f n | ν ,C ≤ ( Cq βn ) q νβ − δνn . Proof of Theorem 1.2
We first introduce some notations. Let p n /q n be the continued fraction expansion of α . Thegeneral settings of the cocycle ( α, A ) are • q n +1 ≤ M q n , n ∈ N , • A ∈ G ν ( S ) with 0 < ν < M, N > ε = M − ≪ δ = 14 M − , λ = e q qNN ≫ , < β < ν . Denote by γ = νβ . For n ≥ N , we inductively defineln λ n +1 = ln λ n − q γ − n +1 , λ N = λ − ε . ln ] λ n +1 = ln f λ n + 10 q γ − n +1 , f λ N = λ ε . LINGRUI GE, YIQIAN WANG, JIANGONG YOU, AND XIN ZHAO
Choose q N sufficiently large such that ∞ P i = N +1 q γ − i < ε , then λ ∞ ≥ λ − ε and f λ ∞ ≤ λ ε .We define • The critical set: C = { c , c } where c ∈ [0 , π ) and c = c + π . • The critical interval: I n, = h c − q βn , c + q βn i , I n, = h c − q βn , c + q βn i and I n = I n, S I n, . • The first return time: For x ∈ I n , we denote the smallest positive integer i with T i x ∈ I n (respectively T − i x ∈ I n ) by r + n ( x ) (respectively r − n ( x )), and define r ± n = min x ∈ I n r ± n ( x ).Obviously, r ± n ≥ q n . • The sample function: The 2 π -periodic smooth function φ is defined assin φ ( x ) = ce − (cid:16) x − c − kπ ) ν + c k +1) π − x ) ν (cid:17) , x ∈ [ c + kπ, c + ( k + 1) π ) , k ∈ Z , where c is sufficiently small. Remark 3.1.
To ensure r ± n ≥ q n , we must require β >
1. To ensure ∞ P i = N +1 q γ − i < ε , we mustrequire βν <
1. Thus our construction is possible only if ν <
1. Indeed, it is essential since if ν >
Remark 3.2.
By (5) in Proposition 2.1 and Corollary 2.2, we have φ ∈ G ν ( S ).For C ≥
1, we denote by I n,i C the sets h c i − Cq βn , c i + Cq βn i , i = 1 , I n C the set I n, C S I n, C .Let Λ = (cid:18) λ λ − (cid:19) . Theorem 1.2 follows from the following two propositions whose proofs will begiven in Section 4.
Proposition 3.1.
There exist functions φ n ( x ) on S ( n = N, N + 1 , · · · ) such that (1) n | φ n − φ n − | ν ,K ≤ λ − qn − n for some K > , if n > N , (2) n A n ( x ) , A n ( T x ) , · · · , A n ( T r + n ( x ) − ( x )) is λ n -hyperbolic for x ∈ I n where A n ( x ) = Λ R π − φ n ( x ) , (3) n It holds ( a ) n s n ( x ) − s ′ n ( x ) = φ ( x ) x ∈ I n , ( b ) n | s n ( x ) − s ′ n ( x ) | ≥ | φ ( x ) | ≥ ce − ν q γn , x ∈ I n \ I n , where s n ( x ) = s ( A r + n n ( x )) , s ′ n ( x ) = s ( A − r − n n ( x )) , (4) n It holds k A r ± n k G
1+ 1 ν ,K ( I n ) ≤ f λ nr ± n , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A r ± n k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G
1+ 1 ν ,K ( I n ) ≤ λ − r ± n n . Proposition 3.2.
There exist functions ˜ φ n ( x ) on S ( n = N, N + 1 , · · · ) such that (1) n | φ n ( x ) − ˜ φ n ( x ) | ν ,K ≤ Cq − n for some K > , if n > N , (2) n e A n ( x ) , e A n ( T x ) , · · · , e A n ( T r + n ( x ) − ( x )) is λ n -hyperbolic for each x ∈ I n where e A n ( x ) = Λ R π − ˜ φ n ( x ) , (3) n It holds ˜ s n ( x ) = ˜ s ′ n ( x ) , x ∈ I n , where ˜ s n ( x ) = s ( e A r + n n ( x )) , ˜ s ′ n ( x ) = s ( e A − r − n n ( x )) . Proof of Theorem 1.2:
By (1)’s in Proposition 3.1 and Proposition 3.2, there exists D ν ∈ G ν ,K such that k A n − D ν k ν ,K → k e A n − D ν k ν ,K →
0. Then Theorem 1.2 is a direct conclusionof the followings:(a) L ( D ν ) ≥ (1 − ε ) ln λ ,(b) L ( e A n ) ≤ (1 − δ ) ln λ, ∀ n > N .Step 1: Proof of (a). We say x ∈ S is nonresonant for A n ( x ) if(3.1) dist ( T i x, C ) > q βN , ≤ i < q N ,dist ( T i x, C ) > q βk , q k − ≤ i < q k , N < k ≤ n. The Lebesgue measure of the set of all nonresonant points x ∈ S is at least 2 π (1 − P N ≤ k 1. For any x satisfying the nonresonant property (3.1),let j be the first time such that T j x ∈ I N and let n be such that T j x ∈ I n \ I n +1 . In general,let j i and n i be defined so that T j i x ∈ I n i \ I n i +1 and let T j i +1 x be the next return of T j i x to I n i . It is obvious that j i +1 − j i ≥ q ni . By condition (2), we have { A n ( T j i x ) , · · · , A n ( T j i +1 − x ) } is λ ∞ -hyperbolic (See also [54] for similar arguments).Since T j i x / ∈ I n i +1 , by (3) n of Proposition 3.1 and the definition of φ , we have ∠ ( s n ( T j i x ) , s ′ n ( T j i x )) ≥ ce − ν q γni +1 . On the other hand, it holds that ∠ (cid:16) s ( A − j i n ( T j i x )) , s ( A j i +1 − j i n ( T j i x )) (cid:17) > ∠ (cid:0) s n ( T j i x ) , s ′ n ( T j i x ) (cid:1) ≥ ce − ν q γni +1 . By Lemma 2.1, we have k A j i +1 n ( x ) k ≥ k A j i n ( x ) k · k A j i +1 − j i n ( T j i x ) k · ∠ ( s ( A − j i n ( T j i x )) , s ( A j i +1 − j i n ( T j i x ))) ≥ k A j i n ( x ) k · λ ( j i +1 − j i )(1 − ε ) ∞ ce − ν q γni +1 . Inductively k A j s n ( x ) k ≥ k A j n ( x ) k · λ j s − j ∞ · s − Y i =0 ce − ν q γni +1 . Notice that j s − j = s P i =1 j i − j i − ≥ s − P i =0 q ni , thus if s is sufficiently large, we have j ≤ q CN ≤ ε j s , s − Y i =0 ce − ν q γni +1 ≥ λ − ε ( j s − j ) . It follows that k A j s n ( x ) k ≥ λ (1 − ε ) j s ∞ ≥ λ (1 − ε ) j s ∞ . Now we are ready to prove the main result. From the subadditivity of the cocycle, the finiteLyapunov exponent of a cocycle converges to the Lyapunov exponent. Thus there exists a large s ≥ N such that (cid:12)(cid:12)(cid:12)(cid:12) j s Z T ln k D j s ν ( x ) k dx − L ( α, D ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. By (1) n of Proposition 3.1, there exists N > N , such that for any n > N , it holds that (cid:12)(cid:12)(cid:12)(cid:12) j s Z T ln k D j s ν ( x ) k dx − j s Z T ln k A j s n ( x ) k dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. On the other hand 1 j s Z T ln k A j s n ( x ) k dx ≥ (1 − ε ) ln λ ∞ − C εC ≥ (1 − ε ) ln λ. Thus we finish the proof of (a).Step 2: Proof of (b). The following two lemmas have been proved in [47]. Lemma 3.1 (Lemma 4.1 of [47]) . Suppose A and B are two hyperbolic matrices such that k A k = λ m and k B k = λ n with m, n > and λ , λ ≫ . If A ( s ( A )) || u ( B ) , then k BA k ≤ { λ m λ − n , λ n λ − m } . Lemma 3.2 (Corollary 4.1 of [47]) . Let min r n = min x ∈ I n min { i > | T i x (mod 2 π ) ∈ I n } , and let max r n = max x ∈ I n min { i > | T i x (mod 2 π ) ∈ I n } , Then M − ≤ min r n max r n ≤ . Let · · · < n j − < n j < n j +1 < · · · be the returning times of x ∈ I n to I n . Moreover, we let n j + be the first returning time of x ∈ I n to I n after n j . Similarly, we denote by n j − the last returningtime of x ∈ I n to I n before n j . Obviously, it holds that n j − ≤ n j − < n j and n j < n j + ≤ n j +1 . ByLemma 3.2, we have n j + − n j , n j − n j − ≤ (1 − M − )( n j + − n j − ) . Since T n j x ∈ I n , by Proposition 3.2 and Lemma 3.1, we have (cid:13)(cid:13)(cid:13) e A n ( T n j + x ) · · · e A n ( T n j x ) · · · e A n ( T n j − x ) (cid:13)(cid:13)(cid:13) ≤ n e A n j + − n j n ( T n j x ) , e A n j − n j − n ( T n j − x ) o ≤ λ max { n j + − n j ,n j − n j − } ≤ λ (1 − M − )( n j + − n j − ) . It follows that (cid:13)(cid:13)(cid:13) e A n ( T n j +1 x ) · · · e A n ( T n j − x ) (cid:13)(cid:13)(cid:13) ≤ λ n j +1 − n j − − M − ( n j + − n j − ) ≤ λ ( n j +1 − n j − )(1 − M − ) . Thus for any even k , (cid:13)(cid:13)(cid:13) e A n ( T n k x ) · · · e A n ( x ) (cid:13)(cid:13)(cid:13) ≤ λ (1 − M − ) P k − j =0 ( n j +2 − n j ) ≤ λ n k (1 − M − ) , which implies that L ( e A n ) ≤ (1 − δ ) ln λ .4. Proof of Proposition 3.1 and 3.2 In this section, we aim to prove Proposition 3.1 and Proposition 3.2 which are the main technicalparts of this paper. The proof is split into the following three subsections.4.1. Key Lemmas. Let 2 < s < ∞ , 0 < γ < K > α ∈ R \ Q be bounded, p n /q n be thecontinued fraction expansion of α and λ > < s < s is sufficiently close to 2.Recall that for any f ∈ C ∞ ( I ), we denote | f | s,K := 4 π k (1 + | k | ) K k ( k !) s | ∂ k f | C ( I ) , the Gevrey norm of f restricting to I .In the following, we prove a Gevrey version of the concatenation of hyperbolic matrices. Itgreatly simplifies and improves the proofs in [47]. Lemma 4.1. Let E ( x ) = (cid:18) e ( x ) 00 e − ( x ) (cid:19) R θ ( x ) (cid:18) e ( x ) 00 e − ( x ) (cid:19) , where e , e , θ ∈ G s,K ( I ) satisfying (4.1) inf x ∈ I (cid:12)(cid:12)(cid:12) θ ( x ) − π (cid:12)(cid:12)(cid:12) ≥ ce − q γn ≫ min (cid:26) inf x ∈ I e ( x ) , inf x ∈ I e ( x ) (cid:27) − , (4.2) (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:12)(cid:12)(cid:12)(cid:12) s,K , | tan θ | s,K ≤ Ce q γn , | cos θ | s,K , | cot θ | s,K ≤ C, (4.3) | e − i | s,K ≤ Cλ − q n − , i = 1 , . Then, for e ( x ) := k E ( x ) k , it holds that (4.4) inf x ∈ I e ( x ) ≥ c inf x ∈ I e ( x ) · inf x ∈ I e ( x ) · e − q γn , (4.5) | e | s, (1+ η ) K ≤ C | e | s,K | e | s,K , (4.6) | e − | s, (1+ η ) K ≤ C | e − | s,K | e − | s,K e q γn . Let s ( x ) = s ( E ( x )) and u ( x ) = u ( E ( x )) , we further have (4.7) (cid:12)(cid:12)(cid:12) π − s (cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ | e − | s,K | e | s,K , | u | s, (1+ η ) K ≤ | e − | s,K | e | s,K , where η = λ − q n − .Proof. For simplicity, let us omit the dependence on x in the following computation. Direct com-putations show that E t E = (cid:18) e e cos θ + e e − sin θ ( e − − e ) sin θ cos θ ( e − − e ) sin θ cos θ e − e − cos θ + e e − sin θ (cid:19) . It is obvious that e + e − = e e cos θ + e e − sin θ + e − e − cos θ + e e − sin θ = e e cos θ (cid:0) e − tan θ + e − e − + e − tan θ (cid:1) := b. Thus(4.8) e = s b + √ b − 42 = √ b s √ − b − , (4.9) e − = s b + √ b − r b s 21 + √ − b − . By (4.1) and (4.8), we have(4.10) inf x ∈ I e ( x ) ≥ c inf x ∈ I e ( x ) · inf x ∈ I e ( x ) · e − q γn . By (1) in Proposition 2.1, (4.3) and (4.2), we have (cid:12)(cid:12) e − tan θ + e − e − + e − tan θ (cid:12)(cid:12) s,K ≤| e − | s,K | tan θ | s,K + | e − | s,K | e − | s,K + | e − | s,K | tan θ | s,K ≤ Cλ − q n − e q γn + Cλ − q n − + Cλ − q n − e q γn ≤ λ − q n − . (4.11)The last inequality holds since α is bounded and λ is sufficiently large. Let η = λ − q n − and η = λ − q n − , by (4.11) and (3)-(4) in Proposition 2.1, we have (cid:12)(cid:12)(cid:12)(cid:12) 11 + e − tan θ + e − e − + e − tan θ − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ λ − q n − . (4.12) (cid:12)(cid:12)(cid:12)(cid:12)q e − tan θ + e − e − + e − tan θ − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ λ − q n − . (4.13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s 11 + e − tan θ + e − e − + e − tan θ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η )(1+ η ) K ≤ λ − q n − . (4.14)By (1) in Proposition 2.1, (4.3), (4.2), (4.12), (4.13) and (4.14), we have (cid:12)(cid:12)(cid:12)(cid:12) b (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ | e − | s,K | e − | s,K (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:12)(cid:12)(cid:12)(cid:12) s,K (cid:12)(cid:12)(cid:12)(cid:12) 11 + e − tan θ + e − e − + e − tan θ (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ C e q γn λ − q n − (1 + λ − q n − ) ≤ λ − q n − . (4.15) (cid:12)(cid:12)(cid:12) √ b (cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ | e | s,K | e | s,K | cos θ | s,K (cid:12)(cid:12)(cid:12)(cid:12)q e − tan θ + e − e − + e − tan θ (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ C | e | s,K | e | s,K . (4.16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η )(1+ η ) K ≤ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:12)(cid:12)(cid:12)(cid:12) s,K (cid:12)(cid:12)(cid:12)(cid:12) e (cid:12)(cid:12)(cid:12)(cid:12) s,K (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:12)(cid:12)(cid:12)(cid:12) s,K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s 11 + e − tan θ + e − e − + e − tan θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η )(1+ η ) K ≤ C | e − | s,K | e − | s,K e q γn . (4.17)By (4.15), (3) and (4) in Proposition 2.1, we have (cid:12)(cid:12)(cid:12)p − b − − (cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ λ − q n − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s √ − b − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η )(1+ η ) K ≤ λ − q n − , (4.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s 21 + √ − b − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η )(1+ η )(1+ η ) K ≤ λ − q n − , (4.19)where η = λ − q n − .By (4.16), (4.17), (4.18) and (4.19), we have | e | s, (1+ η ) K = (cid:12)(cid:12)(cid:12) √ b (cid:12)(cid:12)(cid:12) s, (1+ η ) K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s √ − b − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ C | e | s,K | e | s,K . | e − | s, (1+ η ) K = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s 21 + √ − b − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ C | e − | s,K | e − | s,K e q γn . We finish the proofs of (4.4)-(4.6).Now we prove (4.7). By polar decomposition procedure, we have s ( x ) = π + θ ( E ( x )) where s ( x ) is the most contraction direction of E ( x ) and θ ( E ( x )) is the eigen-direction of E t ( x ) E ( x ) corresponding to the eigenvalue k E ( x ) k . Let a = e e , c = e e , u = 2( e − e − ) sin θ cos θ and U = ( a − a − ) cos θ + ( c − c − ) sin θ . It’s easy to calculate that(4.20) tan s ( x ) = tan( π θ ( E ( x ))) = u √ U + u − U = √ U + u + Uu . Since | θ − π/ | > ce − q γn ≫ min (cid:26) inf x ∈ I e ( x ) , inf x ∈ I e ( x ) (cid:27) − , we have(4.21) U ≥ ce e e − q γn − e e − − e − e > . (4.20) and (4.21) imply that(4.22) s ( x ) = arctan sgn ( u ) r U u + 1 + U | u | !! . A direct calculation shows that g := Uu = ( e e ) − ( e e ) − e − e − ) cot θ + ( e e − ) − ( e e − ) − e − e − ) tan θ. (4.23)Without loss of generality, we only consider the case g ( x ) > 0. A direct computation shows dsdx = 12 11 + g dgdx ,g = e cot θ − e − e − + ( e − − e − ) tan θ − e − := e cot θ · h, 11 + g = 4 e − tan θ e − tan θ + h . By (4.3), (4.2), (1) and (3) in Proposition 2.1, we have | h − | s, (1+ η ) K = (cid:12)(cid:12)(cid:12)(cid:12) − e − e − + ( e − − e − ) tan θ − e − − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ λ − q n − , | h − | s, (1+ η ) K ≤ | h − | s, (1+ η ) K | h + 1 | s, (1+ η ) K ≤ λ − q n − , (cid:12)(cid:12)(cid:12)(cid:12) e − tan θ + h − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ λ − qn − . It follows that | g | s, (1+ η ) K ≤ C | e | s,K | cot θ | s,K (cid:12)(cid:12)(cid:12)(cid:12) − e − e − + ( e − − e − ) tan θ − e − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ C | e | s,K , (cid:12)(cid:12)(cid:12)(cid:12) 11 + g (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ | tan θ | s,K | e − | s,K ≤ C | e − | s,K e q γn . By (2) in Proposition 2.1 and (4.3), we have (cid:12)(cid:12)(cid:12) π − s (cid:12)(cid:12)(cid:12) s, (1+ η ) K ≤ (cid:12)(cid:12)(cid:12)(cid:12) 11 + g (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ η ) K | ∂g | s, (1+ η ) K ≤ C | e − | s,K e q γn | e | s,K Kλ q n − ≤ | e − | s,K | e | s,K . Similar results hold for u . Thus we finish the whole proof. (cid:3) Consider a sequence of maps E ℓ ∈ G s,K ( I, SL (2 , R )) , ≤ ℓ ≤ n − . Let s ℓ ( x ) = s [ E ℓ ( x )], u ℓ ( x ) = u [ E ℓ ( x )], e ℓ ( x ) = k E ℓ ( x ) k and Λ ℓ ( x ) = (cid:18) e ℓ ( x ) 00 ( e ℓ ( x )) − (cid:19) . Bypolar decomposition, it holds that E ℓ ( x ) = R u ℓ ( x ) Λ ℓ ( x ) R π − s ℓ ( x ) . Set for each 0 ≤ ℓ ≤ n − E ℓk ( x ) = E k − ℓ ( x ) · · · E ℓ ( x ) , ≤ k ≤ n − ℓI , k = 0 (cid:16) E ℓ + k − k ( x ) (cid:17) − , − ℓ ≤ k ≤ − . For k ≥ 1, let s ℓk ( x ) = s [ E ℓk ( x )], u ℓk ( x ) = s [ E ℓ − k ( x )], e ℓk ( x ) = k E ℓk ( x ) k and Λ ℓk ( x ) = (cid:18) e ℓk ( x ) 00 ( e ℓk ( x )) − (cid:19) .Again by polar decomposition, it holds that E ℓk ( x ) = R u k + ℓk ( x ) Λ ℓk ( x ) R π − s ℓk ( x ) . Lemma 4.2. Let ≤ ℓ < n − ≤ q Cm , < ξ < and θ ℓ ( x ) =: u ℓ − ( x ) − s ℓ ( x ) + π . Assume that (4.24) inf x ∈ I (cid:12)(cid:12)(cid:12) π − θ ℓ ( x ) (cid:12)(cid:12)(cid:12) = inf x ∈ I (cid:12)(cid:12)(cid:12) s ℓ ( x ) − u ℓ − ( x ) (cid:12)(cid:12)(cid:12) ≥ ce − q γm ≫ min ≤ ℓ ≤ n − (cid:26) inf x ∈ I e ℓ ( x ) (cid:27) − , (4.25) (cid:12)(cid:12)(cid:12)(cid:12) θ ℓ (cid:12)(cid:12)(cid:12)(cid:12) s,K , | tan θ ℓ | s,K ≤ Ce q γm , | cos θ ℓ | s,K , | cot θ ℓ | s,K ≤ C, (4.26) | ( e ℓ ) − | s,K ≤ (cid:16) | e ℓ | s,K (cid:17) − ξ ≤ Cλ − q m − . Then it holds that (4.27) inf x ∈ I e n ( x ) ≥ c n e − nq γm n − Y ℓ =0 inf x ∈ I e ℓ ( x ) , (4.28) | e n | s, (1+ η ) K ≤ C n n − Y ℓ =0 | e ℓ | s,K , (4.29) | ( e n ) − | s, (1+ η ) K ≤ C n n − Y ℓ =0 | ( e ℓ ) − | s,K e nq γm , (4.30) (cid:12)(cid:12) s − s n (cid:12)(cid:12) s, (1+ η ) ≤ λ − q m − , (cid:12)(cid:12) u n − − u nn (cid:12)(cid:12) s, (1+ η ) K ≤ λ − q m − , where η = λ − q m − .Proof. We prove it by induction. In the case of the product of two matrices, i.e. n = 2, it followsfrom Lemma 4.2. Now, we assume for k ≤ n − ℓ , we have(4.31) inf x ∈ I e ℓk ( x ) ≥ c k e − kq γm k − Y j =0 inf x ∈ I e ℓ + j ( x ) , (4.32) | e ℓk | s, (1+ η ) k K ≤ C k k − Y j =0 | e ℓ + j | s,K , (4.33) | ( e ℓk ) − | s, (1+ η ) k K ≤ C k k − Y j =0 | ( e ℓ + j ) − | s,K e kq γm , (4.34) (cid:12)(cid:12)(cid:12) s ℓk − − s ℓk (cid:12)(cid:12)(cid:12) s, (1+ η ) k K ≤ λ − kq m − , (4.35) (cid:12)(cid:12)(cid:12) u n − ℓk − − u n − ℓk (cid:12)(cid:12)(cid:12) s, (1+ η ) k K ≤ λ − kq m − , where η = λ − q m − .Clearly, (4.34) implies that for ℓ = 0 , (cid:12)(cid:12)(cid:12) s ℓ − s ℓn − (cid:12)(cid:12)(cid:12) s, (1+ η ) n − K ≤ n − X j =1 λ − jq m − ≤ λ − q m − , (cid:12)(cid:12)(cid:12) u n − ℓ − − u n − ℓn − (cid:12)(cid:12)(cid:12) s, (1+ η ) n − K ≤ n − X j =1 λ − jq m − ≤ λ − q m − . Combining the above with (4.24), we have(4.36) inf x ∈ I (cid:12)(cid:12) u n − n − ( x ) − s n − ( x ) (cid:12)(cid:12) ≥ ce − q γm , inf x ∈ I (cid:12)(cid:12) s n − ( x ) − u ( x ) (cid:12)(cid:12) ≥ ce − q γm . Let ˜ θ n − ( x ) = u n − n − ( x ) − s n − ( x ), then (cid:12)(cid:12)(cid:12) ˜ θ n − − θ n − (cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ (cid:12)(cid:12) u n − n − − u n (cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ λ − q m − . Note that 1cos ˜ θ n − = 1cos θ n − θ n − − θ n − ) − tan θ n − sin(˜ θ n − − θ n − ) . By (5) in Proposition 2.1 and (4.25), we have (cid:12)(cid:12)(cid:12) cos(˜ θ n − − θ n − ) − tan θ n − sin(˜ θ n − − θ n − ) − (cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ≤ λ − q m − . By (3) in Proposition 2.1, we have(4.37) (cid:12)(cid:12)(cid:12)(cid:12) θ n − − θ n − ) − tan θ n − sin(˜ θ n − − θ n − ) (cid:12)(cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ≤ . By (4.25) and (4.37), we have (cid:12)(cid:12)(cid:12)(cid:12) θ n − (cid:12)(cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) θ n − (cid:12)(cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ Ce q γm . (4.38)Similarly (cid:12)(cid:12)(cid:12) tan ˜ θ n − (cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ Ce q γm , (cid:12)(cid:12)(cid:12) cos ˜ θ n − (cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) , (cid:12)(cid:12)(cid:12) cot ˜ θ n − (cid:12)(cid:12)(cid:12) G s, (1+ η )2 n − K ( I ) ≤ C. (4.39)(4.32), (4.33), (4.36), (4.38) and (4.39) imply that we can apply Lemma 4.2 to the product E n ( x ) = E n − ( x ) E n − ( x ) = E n − ( x ) E ( x ) , which implies thatinf x ∈ I e n ( x ) ≥ c inf x ∈ I e n − ( x ) inf x ∈ I e n − ( x ) e − q γm ≥ c n e − nq γm n − Y ℓ =0 inf x ∈ I e ℓ ( x ) , | e n | s, (1+ η ) n K ≤ C | e n − | s, (1+ η ) n − K | e n − | s, (1+ η ) n − K ≤ C n n − Y ℓ =0 | e ℓ | s,K , | ( e n ) − | s, (1+ η ) n K ≤ C e q γm | ( e n − ) − | s, (1+ η ) n − K | ( e n − ) − | s, (1+ η ) n − K ≤ C n e nq γm n − Y ℓ =0 | ( e ℓ ) − | s,K . By Lemma 4.2 and (4.26), we have (cid:12)(cid:12) s n − − s n (cid:12)(cid:12) s, (1+ η ) n K ≤ | ( e n − ) − | s, (1+ η ) n − K | e n − | s, (1+ η ) n − K ≤ C n − e nq γm n − Y ℓ =0 | ( e ℓ ) − | s, (1+ η ) n − K C n − n − Y ℓ =0 | e ℓ | s, (1+ η ) n − K ≤ C n e nq γm n − Y j =0 | ( e ℓ ) − | s, (1+ η ) n − K ≤ λ − nq m − , thus (cid:12)(cid:12) s − s n (cid:12)(cid:12) s, (1+ η ) n K ≤ (cid:12)(cid:12) s − s n − (cid:12)(cid:12) s, (1+ η ) n K + (cid:12)(cid:12) s n − − s n (cid:12)(cid:12) s, (1+ η ) n K ≤ λ − q m − . Similar results hold for u nn , we finish the proof since (1 + η ) n ≤ λ − q m − for n < q Cm . (cid:3) In the following, we will fix 0 < ν < ν < s = 1 + ν > s = 1 + ν > β > < γ = βν < γ = βν < 1. Let δ > < ν β − δ ν < 1. Recallthat • The critical set: C = { c , c } where c ∈ [0 , π ) and c = c + π . • The critical interval: I n, = h c − q βn , c + q βn i , I n, = h c − q βn , c + q βn i and I n = I n, S I n, . • The first return time: For x ∈ I n , we denote the smallest positive integer i with T i x ∈ I n (respectively T − i x ∈ I n ) by r + n ( x ) (respectively r − n ( x )), and define r ± n = min x ∈ I n r ± n ( x ).Obviously, r ± n ≥ q n . Remark 4.1. If α is bounded, we have r ± n ≤ q Cn for some C only depending on α . See [32] for theproof.4.2. Proof of Proposition 3.1. We prove Proposition 3.1 by induction. Instead of r ± n ( x ), some-times, we use r ± n for short when the difference between r ± n ( x ) and r ± n are negligible. Recall thatln λ n +1 = ln λ n − q γ − n +1 , γ = ν β, λ N = λ − ε . ln ] λ n +1 = ln f λ n + 10 q γ − n +1 , γ = ν β, f λ N = λ ε . We first construct φ N ( x ) and A N ( x ) such that (1) N − (4) N hold. Construction of φ N ( x ) and A N ( x ): Let c , c ∈ T with c ∈ [0 , π ) and c = c + π . We define a2 π -periodic smooth function φ bysin( φ ( x )) = ce − (cid:16) x − c − kπ ) ν + c k +1) π − x ) ν (cid:17) , x ∈ [ c + kπ, c + ( k + 1) π ) , for some 0 < c < . In view of Proposition 2.1 and Corollary 2.2, it’s easy to see(1) φ is a G ν ,C -2 π periodic function for some C > | φ | C ( S ) ≤ π and for i = 1 , | φ ( x ) | ≥ ce −| x − c i | − ν for some c > Let A ( x ) = Λ · R π − φ ( x ) = (cid:18) λ λ − (cid:19) · R π − φ ( x ) , by [54], there exists a large λ > φ , ν and ε such that if λ > λ , { A ( x ) , · · · , A ( T r + N ( x ) − x ) } is λ N − hyperbolic , ∀ x ∈ I N . Let s N ( x ) = s ( A r + N ( x )), s ′ N ( x ) = s ( A − r − N ( x )) for x ∈ I N . Let e N ( x ) be a 2 π -periodic C ∞ -function such that e N ( x ) = φ ( x ) − ( s ′ N ( x ) − s N ( x )) for x ∈ I N . Let ˆ e N ( x ) = e N ( x ) · f N ( x ) where f N is defined in Lemma 2.3 and φ N ( x ) = φ ( x ) + ˆ e N ( x ) for x ∈ S . Verifying (1) N and (4) N of Proposition 3.1 : Let η N = λ − N , n = r + N ≤ q CN , e ℓ ( x ) = k A ( x + ℓα ) k and I = I N , E ℓ ( x ) = A ( x + ℓα ) = Λ · R π − φ ( x + ℓα ) . For 0 ≤ ℓ < n − 1, since x + ℓα / ∈ I N , one can easily verify thatinf x ∈ I k A ( x + ℓα ) k = λ, (4.40) inf x ∈ I (cid:12)(cid:12)(cid:12) π − θ ℓ ( x ) (cid:12)(cid:12)(cid:12) := inf x ∈ I | φ ( x + ℓα ) | ≥ ce − q βν N = ce − q γ N , (4.41) | ( e ℓ ) − | G s ,C ( I ) = | e ℓ | − G s ,C ( I ) = λ − . By Corollary 2.1 and Corollary 2.2, there is some C > | cos θ ℓ | G s ,C ( I ) = | sin( φ ( x + ℓα )) | G s ,C ( I ) , | cot θ ℓ | G s ,C ( I ) = | tan( φ ( x + ℓα )) | G s ,C ( I ) ≤ C, (4.43) (cid:12)(cid:12)(cid:12)(cid:12) θ ℓ (cid:12)(cid:12)(cid:12)(cid:12) G s ,C ( I ) = (cid:12)(cid:12)(cid:12)(cid:12) φ ( x + ℓα )) (cid:12)(cid:12)(cid:12)(cid:12) G s ,C ( I ) , | tan θ ℓ | G s ,C ( I ) = | cot( φ ( x + ℓα )) | G s ,C ( I ) ≤ Ce q γ N . Set q N − = 1, (4.40)-(4.43) imply that all the assumptions in Lemma 4.2 are satisfied. It followsinf x ∈ I N k A r + N ( x ) k ≥ λ r + N c r + N e − r + N q γ N ≥ λ r + N N , k A r + N k G s , (1+ ηN ) C ( I N ) ≤ C r + N λ r + N ≤ f λ N r + N , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A r + N k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G s , (1+ ηN ) C ( I N ) ≤ C r + N λ − r + N e r + N q γ N ≤ λ − r + N N , | e N | G s , (1+ ηN ) C ( I ) ≤ λ − N . Similar results hold for r − N , we omit the proof.By the definition and Lemma 2.3, we have | φ N − φ | G s , (1+ ηN ) C ( S ) = | ˆ e N | G s , (1+ ηN ) C ( S ) ≤ | e N | G s , (1+ ηN ) C ( I N ) | f N | G s , (1+ ηN ) C ( S ) ≤ λ − N ( Cq βN ) q ν β − δ ν N ≤ λ − N . The last inequality holds since ν β − δ ν < λ ≫ e q qNN . Verifying (2) N of Proposition 3.1 : Let A N ( x ) = Λ · R π − φ N ( x ) . Obviously, A N ( x ) = A ( x ) · R − ˆ e N ( x ) . Lemma 4.3 ([47]) . For x ∈ I N , it holds that A r + N N ( x ) = A r + N ( x ) · R − ˆ e N ( x ) and A − r − N N ( x ) = R ˆ e N ( T − r − N x ) · A − r − N ( x ) . Thus, for any x ∈ I N , { A N ( x ) , ..., A N ( T r + N ( x ) − x ) } is a λ N -hyperbolic sequence. Verifying (3) N of Proposition 3.1 : ( s N − s ′ N )( x ) = ( s N − s ′ N )( x ) + ˆ e N ( x ) which implies that( s N − s ′ N )( x ) = φ ( x ) on I N , since | e N ( x ) | ≤ λ − N in I N . Thus we have | ( s N − s ′ N )( x ) | ≥ | φ ( x ) | − λ − N ≥ ce − ν q γ N , on I N \ I N since λ > e q qNN .Inductively, we assume that φ N ( x ) , ..., φ n − ( x ) have been constructed such that Proposition 3.1holds for N ≤ i ≤ n − 1, i.e.,(1) i | φ i ( x ) − φ i − ( x ) | s , (1+ η i ) C ≤ λ − qi − i where η i = Q i − j = N (1 + λ − q j j ) − i For each x ∈ I i , A i ( x ) , A i ( T x ) , · · · , A i ( T r + i ( x ) − ( x )) is λ i -hyperbolic.(3) i We have ( a ) i s i ( x ) − s ′ i ( x ) = φ ( x ) x ∈ I i 10 ;( b ) i | s i ( x ) − s ′ i ( x ) | ≥ | φ ( x ) | ≥ e − ν q ν βi , x ∈ I i \ I i . (4) i It holds k A r ± i k G s , (1+ ηi ) C ( I n ) ≤ e λ ir ± i , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A r ± i k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G s , (1+ ηi ) C ( I n ) ≤ λ − r ± i i . Now we construct φ n ( x ) and verify (1) n − (4) n . Constructing φ n ( x ): From (2) n − , we have that (cid:13)(cid:13)(cid:13)(cid:13) A r + n − ( x ) n − ( x ) (cid:13)(cid:13)(cid:13)(cid:13) · e − (10 q βn − ) ν ≥ λ q n − n − · e − (10 q βn − ) ν ≥ λ (1 − ǫ ) q n − n , x ∈ I n − . Combing the above with (3) n − , for each x ∈ I n , A n − ( x ) , A n − ( T x ) , · · · , A n − ( T r + n ( x ) − ( x )) is λ n -hyperbolic. Let s n ( x ) = s ( A r + n ( x )), s ′ n ( x ) = s ( A − r − n ( x )). Let e n ( x ) be a 2 π -periodic C ∞ -function such that e n ( x ) = φ ( x ) − ( s ′ n ( x ) − s n ( x )) for x ∈ I n . Let ˆ e n ( x ) = e n ( x ) · f n ( x ) and φ n ( x ) = φ ( x ) + ˆ e n ( x ) for x ∈ S . Verifying (1) n and (4) n of Proposition 3.1 : For any x ∈ I n , let j i be defined so that T j i x ∈ I n − \ I n and let T j i +1 x be the next return of T j i x to I n − . Let η n = Q n − j = N (1 + λ − q j j ) − n ≤ q Cn , e ℓ ( x ) = k A j ℓ +1 − j ℓ ( x + j ℓ α ) k and I = I n . E ℓ ( x ) = A j ℓ +1 − j ℓ ( x + j ℓ α ) = R u jℓ +1 − jℓ ( x + j ℓ +1 α ) (cid:18) e ℓ ( x ) 00 ( e ℓ ( x )) − (cid:19) R π − s jℓ +1 − jℓ ( x + j ℓ α ) . By (2) n − , we haveinf x ∈ I k A j ℓ +1 − j ℓ ( x + j ℓ α ) k ≥ λ j ℓ +1 − j ℓ n − ≥ λ qn − n − , ≤ ℓ ≤ n − . By (3) n − , for 0 ≤ ℓ < n − x ∈ I (cid:12)(cid:12)(cid:12) π − θ ℓ ( x ) (cid:12)(cid:12)(cid:12) := (cid:12)(cid:12) s j ℓ +1 − j ℓ ( x + j ℓ α ) − u j ℓ − j ℓ − ( x + j ℓ α ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) s n − ( x + j ℓ α ) − s ′ n − ( x + j ℓ α ) (cid:12)(cid:12) ≥ ce − q βν n = ce − q γ n . By (4) n − , we have | e ℓ | G s, (1+ ηn − C (cid:12)(cid:12)(cid:12)(cid:12) e ℓ (cid:12)(cid:12)(cid:12)(cid:12) G s, (1+ ηn − C ≤ ] λ n − λ n − ! j ℓ +1 − j ℓ ≤ λ ε ( j ℓ +1 − j ℓ ) ≤ | e ℓ | ξG s, (1+ ηn − C , | e ℓ | − ξG s, (1+ ηn − C ≤ λ − jℓ +1 − jℓ n − ≤ λ − qn − n − . By (1) n − , we have | φ n − − φ | G s, (1+ ηn − C ≤ λ − . By Proposition 2.1 and similar arguments asabove, we have | cos θ ℓ | G s , (1+ ηn − C ( I ) , | tan θ ℓ | G s , (1+ ηn − C ( I ) ≤ C, (cid:12)(cid:12)(cid:12)(cid:12) θ ℓ (cid:12)(cid:12)(cid:12)(cid:12) G s , (1+ ηn − C ( I ) , | cot θ | G s , (1+ ηn − C ( I ) ≤ Ce q γ n . Thus all the assumptions in Lemma 4.2 are satisfied, it follows k A r + n k G s , (1+ ηn ) C ≤ C n n − Y ℓ =0 | e ℓ | G s , (1+ ηn − K ≤ C n ] λ n − P n − ℓ =0 ( j ℓ +1 − j ℓ ) ≤ f λ nr + n , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A r + n k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G s , (1+ ηi ) C ≤ C n e n q γn − n − Y ℓ =0 | ( e ℓ ) − | G s , (1+ ηn − K ≤ λ − r + n n , | e n | G s , (1+ ηn ) C ( I n ) ≤ λ − q n − n . Similar results hold for r − n .By the definition, we have | φ n − φ n − | G s , (1+ ηn ) C ( S ) = | ˆ e n | G s , (1+ ηn ) C ( S ) ≤ | e n | G s , (1+ ηn ) C ( I n ) | f n | G s , (1+ ηn ) C ( S ) ≤ λ − qn − n − ( C q βn ) q ν β − δ ν n ≤ λ − q n − n . The last inequality holds because α is bounded. Verifying (2) n of Proposition 3.1 : Define A n ( x ) = Λ · R π − φ n ( x ) . Obviously, A n ( x ) = A n − ( x ) · R − ˆ e n ( x ) . Lemma 4.4 ([47]) . For x ∈ I n , it holds that A r + n n ( x ) = A r + n n − ( x ) · R − ˆ e n ( x ) and A − r − n n ( x ) = R ˆ e n ( T − r − n x ) · A − r − n n − ( x ) . Thus, for any x ∈ I n , { A n ( x ) , ..., A n ( T r + n ( x ) − x ) } is a λ n -hyperbolic sequence. Verifying (3) n of Proposition 3.1 : ( s n − s ′ n )( x ) = ( s n − s ′ n )( x ) + ˆ e n ( x ) which implies that ( s n − s ′ n )( x ) = φ ( x ) on I n , since | e n ( x ) | G s, (1+ ηn ) C ≤ λ − q n − n − in I n . Thus we have | ( s n − s ′ n )( x ) | ≥ | φ ( x ) | − λ − q n − n − ≥ e − (10 q βn ) ν , on I n \ I n .Thus we finish the proof by letting K = lim n →∞ (1 + η n ) C .4.3. Proof of Proposition 3.2. For any n ≥ N , let ˜ e n ( x ) = − ( s n ( x ) − s ′ n ( x )) · f n ( x ) be a 2 π -periodic smooth function such that it is − ( s n ( x ) − s ′ n ( x )) on I n and vanishes outside I n . From (3) n in Proposition 3.1, we have that ˜ e n ( x ) = − φ ( x ) · f n ( x ). Then we define ˜ φ n ( x ) = φ n ( x ) + ˜ e n ( x )and e A n ( x ) = Λ · R π − ˜ φ n ( x ) . Verifying of (1) n of Proposition 3.2 : It follows from the following Lemma. Lemma 4.5. | ˜ e n | G s ,C ≤ e − q γn for n > N .Proof. By (5.7) in Lemma 2.2, we have k φ k G s ,C ( I n ) ≤ e − q βν n for some C > 0. On the otherhand, we choose δ sufficiently small such that βs − − δ = β ν − δ < βν , by Lemma 2.3, we have | f n | s ,C ≤ ( Cq βn ) q βs − − δ n . Thus | ˜ e n | G s ,C ( S ) ≤ | φ | G s ,C ( I n ) | f n | G s ,C ( S ) ≤ e − q βν n ( Cq βn ) q βs − − δ n ≤ e − q γ n . (cid:3) Verifying (2) n of Proposition 3.2 : Since for each x ∈ I n , { A n ( x ) , A n ( T x ) , · · · , A n ( T r + n ( x ) − x ) } is λ n -hyperbolic and ˜ φ n ( x ) = φ n ( x ) on S \ I n , we see that { e A n ( x ) , e A n ( T x ) , · · · , e A n ( T r + n ( x ) − x ) } is λ n -hyperbolic. Thus ˜ s n ( x ) = s ( e A r + n n ( x )) and ˜ s ′ n ( x ) = s ( e A − r + n n ( x )) are well defined. Verifying (3) n of Proposition 3.2 : Notice that ˜ s n ( x ) − ˜ s ′ n ( x ) = s n ( x ) − s ′ n ( x ) − ˜ e n ( x ). Thus fromthe definition of ˜ e n ( x ), it holds that ˜ s n ( x ) = ˜ s ′ n ( x ) x ∈ I n . Thus we finish the whole proof by choosing ν = s − and K = max { K , C } .5. The proofs of technical lemmas Proof of Proposition 2.1. The following two Lemmas will be used frequently. Lemma 5.1 (The Formula of Faa di Bruno, see Theorem 1.3.2 in [38]) . Assume f and g are twosmooth functions in an open interval ( a, b ) , let h = g ◦ f , then h ( n ) ( x ) = X k +2 k + ... + nk n = n n ! k ! k ! ...k n ! g ( k ) ( f ( x )) f (1) ! k f (2) ! k ... f ( n ) n ! ! k n , where k = k + k + ... + k n . Lemma 5.2 (Lemma 1.4.1 in [38]) . X k +2 k + ... + nk n = n k ! k ! k ! ...k n ! R k = R (1 + R ) n − , where k = k + k + ... + k n . By Stirling formula, one has(5.1) (cid:16) ne (cid:17) n ≤ n ! ≤ C (cid:16) ne (cid:17) n √ n. Lemma 5.3. For any < ε ≤ and any σ > , we have n σ ≤ (2 σ ) σ ε − σ (1 + ε ) n . Proof. Note that x σ (1+ ε ) x = e − x ln(1+ ε )+ σ ln x . Let f ( x ) = − x ln(1 + ε ) + σ ln x , then f ′ ( x ) = − ln(1 + ε ) + σx . It follows that max | f ( x ) | = f ( σ ln(1+ ε ) ) = − σ + σ ln σ ln(1+ ε ) . Hence x σ (1 + ε ) x ≤ (cid:18) σ ln(1 + ε ) (cid:19) σ ≤ (cid:18) σε (cid:19) σ . We finish the proof. (cid:3) Proof of Proposition 2.1: The proof of (1) and (2) can be found in [11]. Now we prove (3), notethat (cid:0) x (cid:1) ( n ) = ( − n n ! x n +1 . By Lemma 5.1, we have(5.2) (cid:18) f (cid:19) ( n ) = X k +2 k + ... + nk n = n n ! k ! k ! ...k n ! ( − k k ! f k +1 f (1) ! k f (2) ! k ... f ( n ) n ! ! k n , where k = k + k + ... + k n . Recall that | f | s,K := 4 π n (1 + | n | ) K n ( n !) s | ∂ n f | C ( I ) , it follows that(5.3) inf x ∈ I | f ( x ) | ≥ π (1 − ε ) , sup x ∈ I (cid:12)(cid:12)(cid:12) f ( n ) ( x ) (cid:12)(cid:12)(cid:12) ≤ ( π (1 + ε ) n = 0 ε π K n ( n !) s (1+ n ) n ≥ . Let c = π (1 − ε ). By (5.2) and (5.3), for n ≥ 1, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) f (cid:19) ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k +2 k + ... + nk n = n n ! k ! k ! ...k n ! k ! c k f (1) ! k f (2) ! k ... f ( n ) n ! ! k n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k +2 k + ... + nk n = n n ! k ! k ! ...k n ! k ! c k ε k K n (cid:18) (2!) s (cid:19) k ... (cid:18) ( n !) s n ! (cid:19) k n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By (5.1) and Lemma 5.3, we have n ! ≤ C (cid:16) ne (cid:17) n √ n ≤ C (cid:16) ne (cid:17) n (1 + ε s +1 ) n ε − s +1) . Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (2!) s (cid:19) k ... (cid:18) ( n !) s n ! (cid:19) k n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( s − k ε − k (cid:16) ne (cid:17) ( s − n (1 + ε s +1 ) ( s − n . It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) f (cid:19) ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c − n ! (cid:16) ne (cid:17) ( s − n (1 + ε s +1 ) ( s − n K n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k +2 k + ... + nk n = n k ! k ! k ! ...k n ! ( ε c − C s ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε c − C s ( n !) s h K (1 + ε c − C s )(1 + ε s +1 ) ( s − i n , where k = k + k + ... + k n and the last inequality follows from Lemma 5.2 and (5.1). For sufficientlysmall ε depending on s , we have(1 + ε c − C s )(1 + ε s − ) ( s − ≤ ε s , ε c − C s ≤ ε . Hence by Lemma 5.3 again,4 π n (1 + | n | ) (cid:16) K (1 + ε s +8 ) (cid:17) n ( n !) s (cid:12)(cid:12)(cid:12)(cid:12) ∂ n f (cid:12)(cid:12)(cid:12)(cid:12) C ( I ) ≤ ε | n | (cid:18) ε s +8 (cid:19) − n ≤ ε . By the definition, we have (cid:12)(cid:12)(cid:12)(cid:12) f − (cid:12)(cid:12)(cid:12)(cid:12) s, (1+ ε s +8 ) K ≤ ε . For (4), note that | ( √ x ) n | = | · · · ( − n + 1) x − n | ≤ ( n + 2)! p | x || x | − n . Similar to the proof of(2), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:16)p f (cid:17) ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε c − C s ( n + 2) ( n !) s h K (1 + ε c − C s )(1 + ε s − ) ( s +1) i n . By Lemma 5.3, we have4 π n (1 + | n | ) (cid:16) K (1 + ε s +16 ) (cid:17) n ( n !) s (cid:12)(cid:12)(cid:12) ∂ n p f (cid:12)(cid:12)(cid:12) C ( I ) ≤ ε | n | (cid:18) ε s +16 (cid:19) − n ≤ ε . The proof of (5) is exactly the same as (2) since | arcsin ( n ) x | ≤ n n !, | sin ( n ) x | , | cos ( n ) x | ≤ ≤ n !for any n ∈ N . Thus we finish the proof.5.2. Proof of Lemma 2.2. We inductively prove for x > f ( n ) ( x ) = n X i =1 a ni ( ν ) x iν + n e − xν , (5.5) | a ni ( ν ) | ≤ (2 ν + 2) n + i ( ν + n ) n − i , ≤ i ≤ n. Assume for k ≤ n , (5.4) and (5.5) hold, then for k = n + 1, we have f ( n +1) ( x ) = n X i =1 νa ni ( ν ) x iν + n + ν +1 e − xν − n X i =1 a ni ( ν )( iν + n ) x iν + n +1 e − xν := n +1 X i =1 a n +1 i ( ν ) x iν + n +1 e − xν , where a n +1 i = − a n ( ν )( ν + n ) i = 1 a ni − ( ν ) ν − a ni ( ν )( iν + n ) 2 ≤ i ≤ na nn ( ν ) ν i = n + 1 . By (5.5), we have | a n +1 i | ≤ | a ni ( ν ) | ( ν + n ) ≤ (2 ν + 2) n + i ( ν + n ) n +1 − i ≤ (2 ν + 2) n +1+ i ( ν + n + 1) n +1 − i i = 1 | a ni − ( ν ) ν | + | a ni ( ν )( iν + n ) | ≤ (2 ν + 2) n + i +1 ( ν + n + 1) n +1 − i ≤ i ≤ n | a ni ( ν ) ν | ≤ (2 ν + 2) n +1+ i ( ν + n ) n − i ≤ (2 ν + 2) n +1+ i ( ν + n + 1) n +1 − i i = n + 1 . (5.4) and (5.5) imply that(5.6) | f ( n ) ( x ) | ≤ n X i =1 (2 ν + 2) n + i ( ν + n ) n − i | x | iν + n e − | x | ν . Notice that sup x ∈ R | x | iν + n e − | x | ν ≤ (cid:18) iν + n ) ν (cid:19) i + n/ν , it follows that | f ( n ) ( x ) | ≤ e − | x | ν n X i =1 (2 ν + 2) n + i ( ν + n ) n − i (cid:18) iν + n ) ν (cid:19) i + n/ν ≤ e − | x | ν C n ( ν + n ) n (1+ ν ) ≤ e − | x | ν C n ( n !) ν . (5.7)5.3. Proof of Corollary 2.1. By the same argument as in Lemma 2.2, we have for any x > f ( n ) ( x ) = n X i =1 a ni ( ν ) x iν + n e xν , (5.9) | a ni ( ν ) | ≤ (2 ν + 2) n + i ( ν + n ) n − i , ≤ i ≤ n. (5.8) and (5.9) imply that(5.10) | f ( n ) ( x ) | ≤ n X i =1 (2 ν + 2) n + i ( ν + n ) n − i | x | iν + n e | x | ν . Notice that sup x ∈ R | x | iν + n e − | x | ν ≤ (cid:18) iν + nν (cid:19) i + n/ν , it follows that | f ( n ) ( x ) | ≤ e | x | ν n X i =1 (2 ν + 2) n + i ( ν + n ) n − i (cid:18) iν + nν (cid:19) i + n/ν ≤ e | x | ν C n ( n !) ν . Proof of Corollary 2.2. For any k ∈ Z and x ∈ [ c + kπ, c + ( k + 1) π ), we have g ( x ) = cf ( x − c − kπ ) f ( c + ( k + 1) π − x ) . Let us firstly show that g ∈ C ∞ ( S ), for which we only need to verify the derivative exists for x = c + kπ . By a direct calculation, we have g ( n ) ( x ) = n P k =0 nk ! f ( n − k ) ( x − c − kπ )( − k f ( k ) ( c + ( k + 1) π − x ) c + kπ < x < c + ( k + 1) π n P k =0 nk ! f ( n − k ) ( x − c − ( k − π )( − k f ( k ) ( c + kπ − x ) c + ( k − π < x < c + kπ . We inductively prove that(5.11) g ( n )+ ( c + kπ ) = g ( n ) − ( c + kπ ) = 0 . Assume (5.11) holds for all k ≤ n . For k = n + 1, by Lemma 2.2, we havelim x ց c + kπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( n ) ( x ) − g ( n ) ( c + kπ ) x − c − kπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim x ց c + kπ n P k =0 (cid:18) nk (cid:19) e − | x − c − kπ | ν C n − k (( n − k )!) ν e − | c k +1) π − x | ν C k (( k )!) ν x − c − kπ = 0 . lim x ր c + kπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( n ) ( x ) − g ( n ) ( c + kπ ) x − c − kπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim x ր c + kπ n P k =0 (cid:18) nk (cid:19) e − | x − c − ( k − π | ν C n − k (( n − k )!) ν e − | c kπ − x | ν C k (( k )!) ν | x − c − kπ | = 0 . Thus g ( n +1)+ ( c + kπ ) = g ( n +1) − ( c + kπ ) = 0 . By (1) in Proposition 2.1 and Lemma 2.2, we have | g | s,C ≤ | f ( · − c ) | s,C | f ( c + π − · ) | s,C ≤ ce − | x − c | ν e − | x − c − π | ν . We thus finish the proof.5.5. Proof of Lemma 2.3. Define φ ( x ) = ( e − x /δ x > x ≤ . Let w ( x ) = ( w ( − x ) x > w ( x ) x ≤ , where w ( x ) = φ ( x +2) φ ( x +2)+ φ ( − x − . It’s easy to verify that w ( x ) = x ≥ e − − x +2)1 /δ e − − x +2)1 /δ + e − x − /δ < x < − ≤ x ≤ e − x +2)1 /δ e − x +2)1 /δ + e − − x − /δ − < x < − x ≤ − . By similar arguments as Corollary 2.2, we have w ∈ G δ ( R ).Then we define f n to be a π -periodic function such that f n ( x ) = w (10 q βn ( x − c )) , x ∈ h c − π , c + π i . From the definition, we have that f ( r ) n ( x ) = (10 q n ) βr · w ( r )1 ( y ) where y = 10 q n ( x − c ). By thedefinition of G δ -norm, there exists C > x ∈ I n | f ( r ) n ( x ) | ≤ ( Cq n ) βr ( r !) δ r . Thus sup x ∈ I n | f ( r ) n ( x ) | (1 + r ) C r ( r !) ν ≤ q βrn ( r !) δ − ν ≤ ( Cq βn ) q νβ − δνn . Acknowledgement We would like to thank Svetlana Jitomirskaya for many valuable discussions. L. Ge and X.Zhao were partially supported by NSF DMS-1901462. L. Ge was partially supported by AMS-Simons Travel Grant 2020-2022. Y. 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The H¨older continuity of Lyapunov exponents for a class of Cos-typequasiperiodic Schr¨odinger cocycles. arXiv:2006.03381.[53] J. You and S. Zhang. H¨older continuity of the Lyapunov exponent for analytic quasiperiodicSchr¨odinger cocycle with weak Liouville frequency. Ergod. Th. & Dynam. Sys. (2014),1395-1408.[54] L.S. Young. Lyapunov exponents for some quasi-periodic cocycles. Ergod. Th. & Dynam. Sys. (1997), 483-504. Department of Mathematics, University of California Irvine, CA, 92697-3875, USA Email address : [email protected] Department of Mathematics, Nanjing University, Nanjing 210093, China Email address : [email protected] Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China Email address : [email protected] Department of Mathematics, Nanjing University, Nanjing 210093, China and Department of Math-ematics, University of California Irvine, CA, 92697-3875, USA Email address ::