The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Local Rigidity of Reducibility of Analytic Quasi-periodic Cocycles on U(n) Xuani Hou Jiangong You Vienna, Preprint ESI 1991 (2008) January 3, 2008 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL:

2 Local Rigidity of Reducibility of Analytic Quasi-periodic Cocycles on U(n) Xuani Hou and Jiangong You Department of Mathematics Naning University, Naning ,P.R.China Abstract In this paper, we consider an analytic cocycle (α,a) on T d U(n). We prove that if the cocycle is conugated to a constant cocycle (α,c) by a measurable conugacy (0, B), it is analytic reducible for almost all α and C provided A is close to constant enough. Moreover if B is continuous it is actually analytic. 1 Introduction Let T d = R d /2πZ d be d-dimensional torus. Denote by SW r (T d, G) (r = 0, 1,,, ω) the set of all quasi-periodic cocycles (what is called discrete-time case) on some Lie group G, i.e., diffeommorphism of T d G of the form (α, A) : T d G T d G (x, y) (x + α, A(x)y), where α R d, A C r (T d, G). We will say (α, A) is constant as A is constant. Iterates of (α, A) are of the form (α, A( )) n = (nα, A n ( )), where A 0 I and for n 1 { An ( ) = A( + (n 1)α) A( ) A n ( ) = A( nα) 1 A( α) 1 Cocycles introduced above are important in the study of quasi-periodic linear differential equations of the form { Ẋ = A(θ)X θ = ω where ω R d+1, X : T d+1 G, A : T d+1 g and g is the Lie algebra of G. The flow (ωt, Z t ( )) of it satisfies the cocycle (what is called continuous-time case) property Z t+s ( ) = Z t ( + ωs)z s ( ). The work was partially supported by National Basic Research Program of China (973 Program)(2007CB814800) and NNSF of China (Grant ) 1

3 Since the understanding of the discrete-time case is to great extent enough to understand the continuous-time case, we will focus mainly on the former in this paper. We say that two cocycles (α, A), (α, Ã) SW r (T d, G) are C s or measurable conugated (s r) if there exist B : T d G be C s differentiable or ust measurable such that Ad(B).(α, A) := (α, B( + α)ab 1 ) = (α, Ã). If (α, A) can be C s or measurable conugated to a constant we say it is C s or measurable reducible correspondingly. Note A will homotopic to constant map provided (α, A) is continuous reducible. The dynamic of a reducible cocycle (α, A) SW r (T d, G) is well understood since (α, A) n = (nα, A n ) = (nα, (B +α)c n B 1 ), for some B : T d G and constant C G. Usually conugacy with high differentiability carries more information than low ones. For example, if B is C r differentiable all k th (1 k r or 1 k < as r =, ω) derivatives of A n will be uniformly bounded. Moreover, if all eigenvalues of C are in {z C : z = 1}, {A n : n Z} will be precompact with respect to C r topology. In general low differentiable reducibility of a cocycle is not always imply high differentiable reducibility even the cocycle we consider is smooth enough. A nature question is when a C r cocycle has measurable or C 0 reducibility will have C r reducibility. Some arithmetic condition will be necessary when considering such rigidity problem. To see this, let us check the reducibility of cocycle (α, R ϕ(x) ) SW r (T d, SO(2, R)) where R θ indicates the rotation of angle θ and ϕ C r (T d, R). The reducibility amounts to find a solution f of the equation f(x + α) f(x) = ϕ(x) ϕ(θ)dθ T d Sometimes, the above equation will have continuous while not smooth solution. However, as α DC(γ, τ), we can find solution f C r d 1 τ (or C r as r =, ω). Here we say α T d satisfies a Diophantine condition DC(γ, τ), γ, τ > 0 if e 1<k,α> 1 > γ 1 k τ, 0 k Zd. Let DC = γ,τ>0 DC(γ, τ). Note γ>0 DC(γ, τ) has full Haar measure provided τ > d 1. It is well known that quasi-periodic cocycles on SL(2, R) has rotation number (see [JM82]). Eliasson has proved in [El92] that for α be Diophantine, any (α, A) with A : T d SL(2, R) be C r (r =, ω) close enough to constants is C r reducible provided the rotation number ρ = 1 2 < k, α > for some k Zd or Diophantine with respect to α, that is e 1<k,α> e 2ρ 1 > χ (1 + k ) ν, k Zd, χ, ν > 0. But for quasi-periodic cocycles on Lie group other than SL(2, R), there is no unique and independent way to define something like rotation number. Nevertheless, when a C r cocycle (α, A) continuous or measurable conugate to some constant (α, C), one can 2

4 consider the eigenvalues of C. We want to make clear the relations between the arithmetic conditions of the eigenvalues of C and the C r reducibility of (α, A), particularly in the case A is close to constants. It is another reason why we are interested in the rigidity of reducibility. Notice we here consider only one cocycle other than a family as in the full measure reducibility problem (one can refer to [El92, El98, Kr99a, Kr99b, HY04, etc.] for the describer and results of such problem). Remark 1.1 The original result of Eliasson in [El92] is for the case of r = ω and quasiperiodic linear differential equations, but the proof for the case of discrete-time cocycles or r = is ust the same. In this paper, we will focus on analytic quasi-periodic cocycles on T d U(n) close to constants with respect to C ω topology and prove a local rigidity result of reducibility. In a forthcoming paper, we will use Krikorian s renormalization scheme (as in [Kr01]) together with the local result obtained in this paper to prove a global rigidity result. Recall U(n) is composite of all A GL(n, C) satisfy A A = I, where A denotes the conugate transposition of A and I denotes the identity. The corresponding Lie algebra u(n) is composite of all X gl(n, C) satisfy X + X = 0. Notice any A U(n) is diagonalizable, and eigenvalues spec(a) {z C : z = 1}. Let Σ α = χ>0,ν>0 Σ α(χ, ν), where Σ α (χ, ν) denote the set of all A U(n), such that for any λ µ spec(a) e 1<k,α> µ λ χ (1 + k ) ν, k Zd. Denote by C ω h (Td, M) the set of all map A : T d M which has a holomorphic extension in a complex neighborhood of T d with radius h, that is T d h := {(z 1,,z d ) C d : Imz i < h, i = 1,,d}/2πZ d, and where M can be a Lie group or a Lie algebra. Note C ω h (Td, g) is a Banach space for any Lie algebra g, with the norm F h := sup F(x). x T d h The following is the main theorem of this paper. Theorem 1.1 Assume α DC(γ, τ), (α, Ae F ) SW ω (T d, U(n)) with F C ω h (Td, u(n)) (h > 0), and there exists Λ Σ α and B : T d U(n) measurable such that Ad(B).(α, Ae F ) = (α,λ). Then there exist δ = δ(d, h, γ, τ, B) > 0, if F h < δ, one can find B C ω (T d, U(n)) such that B(x) = B(x) for a.e. x T d and then Ad( B).(Ae F ) = (α,λ). If B is continuous it is analytic actually. Remark 1.2 Similar result can be expected for cocycles in other groups such as GL(n, R), GL(n, C). The proof would be technically more difficult since the matrices are not always diagonalizable. Corollary 1.1 In above theorem 1.1, if we use any compact Lie subgroup G (SU(n), SO(n, R), ect.) of U(n) instead, the corresponding conclusions will also be true. 3

5 Proof: If we can find B C ω (T d, U(n)) such that Ad( B).(α, Ae F ) = (α,λ) with B(x) = B(x) for a.e. x T d. Thus B(x) G for a.e. x T d, so B C ω (T d, G). Idea of proof of main theorem: Similar to Eliasson [El92], we first give a formal reducibility result, then prove the conugacy is actually analytic for any Λ Σ α (provided α be Diophantine). More precisely, we construct a Y C ω (T d, U(n)) such that Ad(e Y ).(α, Ae F ) = (α, A + e F + ), with F + much smaller than F provided F ε is small enough. If spec(a) = {λ 1,,λ n } satisfies e 1<k,α> λ t λ s ǫ, k Z d, s t {1,,n}, for some ǫ > 0 not too small, one can take ǫ ε σ for some σ (0, 1). We then can solves the linearized equation A Y (. + α)a Y = F + A A +, to find A + and Y and then simply defines F + by e F + = A +e Y (.+α) Ae F e Y. Then F + ε 1+σ. If the above property of spec(a) does not hold, we can find Q C ω (T d, U(n)) (not close to identity), such that Ad(Q).A const satisfies above assumption. Then we use above argument to find A + and Y, such that F + ε 1+σ, where e F + = A +e Y (.+α) Q(. + α)ae F Q 1 e Y. Repeat the above procedure, we obtain a sequence of cocycle (α, A e F ) with F converges to 0 very fast and a sequence of conugacy R C ω (T d, U(n)) conugate (α, A e F ) to (α, A +1 e F +1 ). If =1 R is C ω convergent to some analytic map then Ae F can be analytical reducible. There is a problem that when the R not close to identity occurs infinitely times, we have no way to prove the convergence. Fortunately, one can prove it is not the case provided Λ Σ α and F is small enough. 2 Formal reducibility In this section we shall construct a series change of variables such that the composition of them conugates the cocycle (α, Ae F ) to a constant. Those change of variables have two different types, one is close to identity and another one are used to cancel the resonances among the eigenvalues of A. The idea has been used to due with cocycles on SL(2, R), GL(n, R), SO(3, R) by Eliasson in [El92, El01, El02] and to due with cocycles on general semi-simple compact group by Krikorian in [Kr99a, Kr99b]. Here we repeat the proof since we need the estimates to get analytic reducibility. We will leave the proof of convergence of the composition to the next section. 4

6 Notice F C ω h (Td, gl(n, C)) has Fourier expansion F(x) = k Z d F(k)e 1<k,x>, with the estimates F(k) F h e k h. And any F C ω h (Td, gl(n, C)) is in C ω h (Td, u(n)) if and only if F(k) = F( k). For any N N we use the notation T N F and R N F denote k N F(k)e 1<k,x> and k >N F(k)e 1<k,x>. We will say a subset Υ of {z C : z = 1} is (N, ǫ) non-resonant if for any λ, µ Υ e 1<k,α> µ λ ǫ for all k Z d with 0 k N. Otherwise we say it is (N, ǫ) resonant. We say A U(n) is (N, ǫ) resonant (non-resonant) if spec(a) is (N, ǫ) resonant (non-resonant). 2.1 One Step of KAM. We want to solve the linearized homological equation A Y (. + α)a Y = F + F(0). (2. 1) For T N F will very close to F as N large enough (the difference is R N F), we mostly ust need to solve the equation A Y (. + α)a Y = T N F F(0). (2. 2) There exists constant c 1, such that for any N N and δ > 0 we have 0 k N e k δ c 1 δ d and then obtain for any N N and k >N e k δ c 1 N d e Nδ δ d, R N F h+ k >N e k (h h +) F h c 1 N d e N(h h +) (h h + ) d F h. (2. 3) The following small divisor lemma give the estimates for the solution of (2.2). Lemma 2.1 (Small Divisor Lemma) Assume A U(n) is (N, ǫ) non-resonant. Then for any F C ω h (Td, u(n)), (2.2) has a solution Y C ω h + (T d, u(n)) (0 < h + < h) with Y h+ c 1 ǫ(h h + ) d F h. (2. 4) 5

7 Proof: Without lose of generality, we assume A = diag(λ 1,,λ n ). Let Ŷ (k) = 0 for k = 0 or k > N and for 0 < k N, s, t = 1,,n Ŷ s,t (k) = F s,t (k) λ s λ t e 1<k,α> 1. It s easy to see Ŷs,t(k) = Ŷt,s( k) (k Z d, s, t = 1,,n). And then Y (x) = Y s,t (k)e 1<k,x> k Z d is in C ω h + (T d, u(n)) (h > h + > 0) with the estimate Y h+ k Z d Ŷ (k) e k h+ ǫ 0 k N F h e k (h h +) c 1 F h ǫ(h h + ) d. Given (α, Ae F ) SW ω (T d, U(n)) and Y C ω (T d, u(n)), one can conugated (α, Ae F ) and (α, A + P) by e Y provided P = A +e Y (.+α) Ae F e Y. As Y is the solution of (2.2), the following lemma give the estimates for P I with A + been chose specially, and the proof is ust simple computation. Lemma 2.2 Assume A U(n), F C ω h (Td, u(n)) and Y C ω h + (T d, u(n)) (h > h + > 0) satisfies (2.2). Let A + = Ae F(0) and P = A +e Y (.+α) Ae F e Y, if F h, Y h+ 1, we have for some universal constant c 2 the estimate P I h+ c 2 ( F 2 h + F h Y h+ + Y 2 h + + R N F h+ ). (2. 5) The following is true by implicit function theorem (refer to Appendix for details). Lemma 2.3 There exists η > 0, for any h > 0 and any P C ω h (Td, U(n)) satisfies P I h < η, one can find F + C ω h (Td, u(n)) such that P = e F + and F + h 2 P I h. 2.2 Removing Resonances. The small divisor lemma holds only when spec(a) is (N, ǫ) non-resonant. When spec(a) is (N, ǫ) resonant, we have to remove the resonances first. The original idea of removing resonance is given by Moser and Pöschel in [MP84] and then used by Eliasson in [El92] to obtain his famous results. The following lemma and the proof is due to [El01]. Lemma 2.4 Let α DC(γ, τ). If {λ 1,,λ n } is (N, ǫ) resonant, there exists p 4 n, such that {λ 1 e 1<k 1,α>,,λ n e 1<k 1,α> } is (3pN, ǫ) non-resonant for some k 1,,k n Z d with k 1,, k n pn, provided ǫ < (γ2 n 3 τ 4 nτ N τ ) 1. Proof: For any discrete subset Υ of {z C : z = 1}, define Υ = max{ λ µ : λ, µ Υ}. If Υ < γ 1 N τ ǫ 6

8 then Υ is (N, ǫ) non-resonant. If Λ := {λ 1,,λ n } is resonant, λ i, λ and k Z d with 0 < k N, such that e 1<k,α> λ λ i < ǫ. Then let p 1 = 1, ι 1 = ǫ. From Λ we get n 1 sets with Λ 1 1 = {λ i, e 1<k,α> λ }, and Λ 1 2,,Λ1 n 1 denote the sets of one element in Λ/Λ1 1. Note Λ1 s (1 s n 1) is (3p 1 N, ǫ) non-resonant for Λ 1 s < ι 1 < γ 1 (3p 1 N) τ ǫ. If Λ 1 1 Λ1 n 1 is (3p 1N, ǫ) non-resonant, we have proved the lemma. If not, λ i Λ i, λ Λ for some i, and k Z d with 0 < k 3p 1 N, such that e 1<k,α> λ λ i < ǫ. Then let p 2 = (3 + 1)p 1 = 4p 1, ι 2 = 2ι 1 + ǫ, Λ 2 1 = Λ1 i (e 1<k,α> Λ 1 ) and the remainder sets denote by Λ 2 2,,Λ2 n 2. Any Λ2 s (1 s n 2) is (3p 2 N, ǫ) non-resonant for Λ 2 s < ι 2 < γ 1 (3p 2 N) τ ǫ. Repeat above process, until for some r < n 1, Λ r 1 Λr n r is (3p r N, ǫ) non-resonant, where p r = 4 r 1, or at n 1 step we obtain one set Λ1 n 1 of n elements which is (3p n 1 N, ǫ) for Λ1 n 1 < ι n 1 < γ 1 (3p n 1 N) τ ǫ,where p n 1 = 4 n 2 N, ι n 1 = (2 n )ǫ = (2 n 1)ǫ. 2.3 Iterations. Now we are in the position to construct a series change of variables which makes the perturbation converges to zero. We begin with a system (α, A 1 e F 1 ) = (α, Ae F ) with F C ω h (Td, U(n)) for some h = h 1 > 0. Let ε 1 = F h. Assume for r = 2,, there exists R r 1 C ω h r (T d, U(n)) and (α, A r e Fr ) SW ω (T d, U(n)), such that Ad(R r 1 ).(A r 1 e F r 1 ) = A r e Fr, where F r C ω h r (T d, U(n)) with F r hr < ε (1+σ)r 1 1 for some fixed small positive number σ (for example σ (0, 4 (n+5) ) will satisfies our needs) and a decreasing sequence of h r > 0 satisfy h r h 1 4 r and h r 1 h r h 1 2 3r. Let ε = ε (1+σ) 1 1, c = max{c 1, c 2, η}. We will prove in the following that if ε satisfies ε < min{(8c) 2/σ, c}, (2. 6) ε σ ε σ 2 we can find R C ω h +1 (T d, U(n)) such that c( 23(+1) h 1 ) d, (2. 7) ce( 26(+1) 4 n σ h 2 1 ln 1 ε ) d, (2. 8) ε σ > γ4 (n+1)τ [ 4+2 h 1 σ lnε ] τ, (2. 9) Ad(R ).(A e F ) = A +1 e F +1 7

9 for F +1 Ch ω +1 (T d, U(n)) with F +1 h+1 < ε +1. It is not difficult to see there exists δ 1 = δ 1 (d, h, σ, γ, τ) > 0, such that if ε 1 < δ 1, all ε ( = 1, 2, ) will satisfy these conditions. Denote by NR(N, ǫ) (RS(N, ǫ)) the set of matrixes in U(n) which is (N, ǫ) non-resonant (resonant). Let N = [ 2+3 σ ln 1 ] and h ε N = [ 8σ ln 1 ]. h ε We then can prove the following Lemma. Lemma 2.5 Assume ε satisfy (2.6, 2.7, 2.8, 2.9), let h +1 = 1 4 h when A NR(N, ε σ ), (2. 10) h +1 = ( )h when A RS(N, ε σ ). (2. 11) There exists A +1 U(n), F +1 C ω h +1 (T d, u(n)) with F +1 h+1 < ε +1 and R C ω h +1 (T d, U(n)), verifying Ad(R ).(α, A e F ) = (α, A +1 e F +1 ). (2. 12) And R = e Y for Y C ω h +1 (T d, u(n)) with Y h+1 < ε 1 2σ as A NR(N, ε σ ). Proof: Note h +1 h 1 /4 +1 and h h +1 1/2 3(+1). Case a): A NR(N, ε σ ). We can find Y C ω h +1 (T d, u(n)) satisfy Moreover, we have the estimates A Y (. + α)a Y = T N F + F (0). R N F h+1 ce( 2(6+5) σ h 2 1 ln 1 ε ) d ε 1+2σ ε 1+3σ/2, (2. 13) Y h+1 c 2(3+2)d h d ε 1 σ ε 1 2σ (2. 14) 1 Let A +1 = A e F (0) and P +1 = A +1 ey (.+α) A e F e Y, satisfy P +1 I h+1 c(ε 1 σ + ε 1 2σ + ε 1 3σ + ε σ 2 )ε 1+σ (2. 15) 1 2 ε1+σ. (2. 16) Thus there exists F +1 C ω h +1 (T d, u(n)) such that e F +1 = P +1 with F +1 h+1 2 P +1 I h+1 < ε 1+σ. (2. 17) Case b): A RS(N, ε σ ) but A NR( N, ε σ ). The proof and the estimates of this case is ust the same as the proof of case a) except the choice of h +1 is different. 8

10 Case c): A RS( N, ε σ ). There exist S U(n), such that S A S = diag(λ 1,,λ n). From (2.8) we obtain ε σ 1 < γ2 n 3 τ 4 nτ N τ Then by Lemma 2.4, there exists p 4 n, and. (2. 18) with k 1,, k n p N, such that Q (x) = diag(e <k 1,x> 1,,e <kn,x> 1 ) (2. 19) C := Q (. + α)s A S Q (.) 1 = diag(λ 1 e<k 1,α> 1,,λ ne <kn,α> 1 ) (2. 20) is (3p N, ε σ ) non-resonant. Let G = Q S F S Q 1, satisfies (where K = 4 n ) and G h+1 e 4p N h +1 F h e K ε 1 8Kσ R N G c( 4(+1) K N ) d e Nh 4 ε 3h 1 ce( 2(4+5) Kσ h 2 1 ln 1 ε ) d ε 1+2σ ε 1 (8K+1)σ, (2. 21) ε 1+3σ/2. (2. 22) We can find Z C ω h +1 (T d, u(n)) satisfies with the estimates A Z (. + α)a Z = T N G + Ĝ(0), Z h+1 c 4(+1)d h d ε 1 (8K+1)σ ε 1 (8K+2)σ. (2. 23) 1 Let A +1 = C eĝ(0) and P +1 = A +1 ez (.+α) C e G e Z, satisfies the estimates P +1 I h+1 c(ε 2 (16K+4)σ c(ε 1 (16K+5)σ + ε 2 (16K+3)σ + ε 2 (16K+2)σ + ε 1+3σ/2 ) (2. 24) + ε 1 (16K+4)σ + ε 1 (16K+3)σ + ε σ 2 )ε 1+σ (2. 25) 1 2 ε1+σ. (2. 26) Thus there exists F +1 C ω h +1 (T d, u(n)) such that P +1 = e F +1 with F +1 h+1 2 P +1 I h+1 ε 1+σ. (2. 27) 9

11 3 Analytic Reducibility We have finished the construction of a formal conugacy R ( ) = =1 R which conugate (α, A 1 e F 1 ) = (α, Ae F ) to some constant cocycle. In this section we shall prove that R ( ) is actually analytic provided (α, A 1 e F 1 ) is measurably conugates to a constant cocycle which satisfies some generic arithmetic condition and F h small enough. Assume now (without lose of generality) there exists {Λ 1,,Λ n } {z C : z = 1}, such that for some B 1 : T d U(n) measurable Ad(B 1 ).(α, Ae F ) = (α,λ) = (α, diag(λ 1,,Λ n )). (3. 1) For measurable B : T d gl(n, C), let B s,t (k) the (s, t) component of it s k th Fourier coefficients B(k), and define [B] := inf 1 s n sup{ B s,t (k) : 1 t n, k Z d }. (3. 2) One can see [B] > 0 for any measurable B : T d GL(n, C). Denote by Γ(N, ǫ) the set of measurable B : T d gl(n, C) satisfies [T N B] ǫ, where T N (B)(x) = k N B(k)e 1<k,x>. We need the following lemma for the next proof. Lemma 3.1 For any N N and ǫ, the following are true. a)γ(n, ǫ)s Γ(N, ǫ/n) where S U(n); b)γ(n, ǫ)e 1diag(<k 1,x>,...,<k n,x>) Γ(N + Ñ, ǫ) where k 1,, k n Ñ; c)γ(n, ǫ)(i + P) Γ(N, ǫ ) where P C 0 (T d, gl(n, C)) and = sup x T d P(x). Proof: Choose any B Γ(N, ǫ). a)denote by B 1 (k),, B n (k) the row vectors of B(k). Let S 1,,S n the row vectors of S and they are pairwise orthogonal. Let C = BS, then we have max Ĉs,t(k) = max < B s (k), St > B s (k) 1 t n 1 t n n B s,t (k), (3. 3) n for all s {1,,n} and k Z d. So C Γ(N, ǫ/n). b) Let C(x) = B(x)exp{ 1diag(< k 1, x >,...,< k n, x >)}. Then there exist k = k(s, t) for any s, t {1,,n} with k(s, t) Ñ, such that Ĉs,t(k + k(s, t)) = B s,t (k) for any k Z d. So [T C] [T N+Ñ NB] ǫ. c)let = BP. Note s,t (k) (k) 1 (2π) d for any s, t {1,,n} and k Z d. B(x)P(x)e 1<k,x> dx, (3. 4) T d For above B 1 : T d U(n) measurable, there exists L 1 N such that B 1 Γ(L 1, [B 1 ]/2). Define for > 1 L = L n (N N 1 ). We will always assume α DC(γ, τ), 0 < σ < 4 (n+4) and in the following. ε 1 < δ = δ(d, h 1, σ, γ, τ,[b 1 ]) := min{δ 1 (d, h 1, σ, γ, τ), ([B 1 ]/2) 1/σ, 2 1/σ2 } 10

12 Lemma 3.2 There exists B Γ(L, ε σ ), such that for = 1, 2, Ad(B ).(α, A e F ) = (α,λ). Proof: It s true for = 1, we will prove it s true for + 1 provided it s true for. If A is (N, ε σ ) or ( N, ε σ ) non-resonant, there exist Y C ω h (T d, u(n)) with such that Y h+1 < ε 1 2σ Ad(e Y ).(α, A e F ) = (α, A +1 e F +1 ). (3. 5) Let B +1 = B e Y, note 1 3σ σ 2 and ε σ2 ε σ , thus ε σ ε 1 2σ So B +1 Γ(L, ε σ ε1 2σ ) Γ(L +1, ε σ +1 ). If A is ( N, ε σ ) resonant, there exists S U(n), = ε σ (1 ε 1 3σ ) ε σ (1 ε σ2 ) ε σ(1+σ). (3. 6) Q (x) = exp{ 1diag{< k 1, x >,,< k n, x >}}, (3. 7) and Z C ω h +1 (Td, u(n)) with Z h < ε 1 (4n+2 +2) such that Ad(e Z Q S ).(α, A e F ) = (α, A +1 e F +1 ). (3. 8) Let B +1 = B S 1 Q 1 e Z, note 1 (4 n+2 + 3)σ σ 2 and ε σ2 ε σ n, thus 1 n εσ ε 1 (4n+2 +2)σ So B +1 Γ(L + 4 n N, 1 n εσ ε1 2σ ) Γ(L +1, ε σ +1 ). = ε σ ( 1 n +3)σ ε1 (4n+2 ) ε σ ( 1 n εσ2 ) ε σ(1+σ). (3. 9) Lemma 3.3 If Λ Σ α, there exists h > 0 s.t. R ( ) = =1 R C ω h (T d, U(n)). Proof: Λ Σ α, χ, ν > 0, s.t. for any r s and 0 k Z d Λ r e 1<k,α> Λ s Let C = B S Γ(N n (N N 1 ), 1 n εσ ), then and then we have χ. (3. 10) (1 + k ) ν C (. + α)diag(λ () 1,,λ() n )e S F S = diag(λ 1,,Λ n )C. (3. 11) where = C (. + α)(e S F S I). e 1<k,α> λ () t Ĉ,s,t (k) +,s,t (k) = Λ sĉ,s,t(k), (3. 12) 11

13 For,s,t (k) h cε, so for any s, t {1,,n}, k Z d Ĉ(k),s,t λ () t e 1<k,α> Λ s cε. (3. 13) s, k Z d with k < L, and t {1,,n}, such that Ĉ,s,t(k) ε σ /n, and thus λ () t e 1<k,α> Λ s ncε 1 σ. (3. 14) Moreover, as large enough, for s 1 s 2, corresponding t 1 and t 2 (with some k 1 and k 2 ) satisfy above equality must be different for the reason as ε small enough λ () t 1 λ () t 2 Λ s1 e 1< k 1 +k 2,α> Λ s2 2ncε 1 σ χ (2L + 1) ν 2ncε1 σ > 0. (3. 15) We want to show A will be (N, ε σ ) non-resonant as is large enough. In fact, if A is (N, ε σ ) resonant, k Zd with 0 < k N, and r, s {1, n}, s.t. λ r e 1<k,α> λ s ε σ. If r = s, we have ε σ e 1<k,α> 1 γ 1 N τ γ 1 h τ 1 2 3(+1)τ (ln 1. (3. 16) ε ) τ It will not be true as ε is small enough. If r s, as is large enough, there exists Λ(r) Λ(s) {Λ 1,,Λ n } and k(r), k(s) Z d with k(r), k(s) L, such that λ r e 1<k(r),α> Λ(r), λ s e 1<k(s),α> Λ(s) ncε 1 σ. (3. 17) Then let l = k k(r) + k(s) and we have Λ(r)e 1<l,α> Λ(s) = Λ(r)e 1<k k(r),α> Λ(s)e 1< k(s),α> (3. 18) On the other hand Λ(r)e 1<l,α> Λ(s) λ(r)e 1<k,α> λ(s) + 2ncε 1 σ (3. 19) ε σ + 2ncε 1 σ. (3. 20) χ ( l + 1) ν χ. (3. 21) (N + 2L + 1) ν So we obtain χ (N + 2L + 1) ν εσ + 2ncε 1 σ. (3. 22) But it is not true as large enough. Now there exists 0, when 0, h +1 = (1 1/2 +2 )h and R = e Y Ch ω (T d, u(n)) with Y < ε 1 2σ. Note for Y = 0 (1 1/2 +2 ) > 0, and 12 (1 + ε 1 2σ ) <. = 0

14 So =1 R converge uniformly to some analytic function in T d h where h = [ (1 1/2 +2 )]h 0. = 0 Lemma 3.4 Assume B : T d U(n) measurable satisfies Ad(B).(α, C) = (α, D) with C = U 1 diag(λ (1) 1,,Λ(1) n )U 1 and D = U 2 disg(λ (2) 1,,Λ(2) n )U 2 for some U 1, U 2 U(n). Then there exists k 1,,k n Z d such that Λ (2) 1 = Λ (1) 1 e 1<k 1,α>,,Λ (2) n = Λ (1) n e 1<k 1,α>. Moreover, if C Σ α, we have for a.e. x T d B(x) = U 1 exp{ 1diag(< k 1, x >,,< k n, x >)}U 2. Proof: Let B = U 2 BU 1. We then have B(. + α)diag(λ (1) 1,,Λ(1) n ) = diag(λ (2) 1,,Λ(2) n ) B. (3. 23) Thus for any s, t {1,,n} and k Z d Λ (1) t B s,t (. + α) = Λ (2) s (e 1<k,α> Λ (1) t Λ (2) s ) B s,t (k) = 0. So there exists k 1,,k n Z d such that Λ (2) 1 = Λ (1) 1 e 1<k 1,α>,,Λ (2) n = Λ (1) n e 1<k 1,α>. Now let E(x) = diag(e 1<k 1,x>,,e 1<k n,x> ) B(x), then B s,t. And we then obtain E(. + α)diag(λ (1) 1,,Λ(1) n ) = diag(λ (1) 1,,Λ(1) n )E. (3. 24) And then for all s, t {1,,n} and k Z d, (e 1<k,α> Λ (1) t Λ (1) s )Ês,t(k) = 0. Note e 1<k,α> Λ (1) t Λ (1) s 0 as k 0, so Ê(k) = 0 for all 0 k Zd. We then can prove the theorem 1.1 by the lemma 2.5, 3.3, 3.4. proof of theorem 1.1: Assume α DC(γ, τ), Λ Σ α and F h1 < δ = δ(d, h 1, σ, γ, τ, ǫ 0 ). We then can find R such that R ( ) = =1 R C ω (T d, U(n)) satisfies for some constant Λ U(n). Thus we have Ad(R ( ) ).(α, Ae F ) = (α, Λ) (3. 25) Ad(R ( ) B ).(α,λ) = (α, Λ) (3. 26) By lemma 3.4, there exists U 1, U 2 U(n) and k 1,,k n Z d, such that (R ( ) B 1)(x) := U 1 exp{ 1diag(< k 1, x >,,< k n, x >)}U 2 (3. 27) for a.e. x T d. That is to say B(x) := U 2exp{ 1diag(< k 1, x >,,< k n, x >)}U 1R ( ) (x) = B 1 (x) (3. 28) for a.e. x T d. We then obtain B C ω (T d, U(n)) with Ad( B).(α, Ae F ) Ad(B).(α, Ae F ) = (α,λ). (3. 29) Moreover, B itself will be analytic provided it is continuous. 13

15 4 Appendix In this section, we prove the fact that for any P C ω h (Td, U(n)) (h > 0) close uniformly in Π h to I enough (the closeness does not depend on h), we can find F C ω h (Td, u(n)) such that P = e F. The Lemmas and proofs here are all standard (one can refer to [Sc]), with some small motivation for our use. Lemma 4.1 (Contracting Mapping Principle) Let X be a complete metric space and f : U X, U open in X, and assume d(f(x), f(y)) d(x, y) with 0 < 1, where d(x, y) is the distance between x and y. Moreover, suppose there exists z 0 such that d(z 0, X U) > M, and d(z 0, φ(z 0 )) < M(1 ). Then there exists a fixed z = φ(z ) such that d(z 0, z ) < M. Proof: d(z 0, φ(z 0 )) < M(1 ) < M < d(z 0, X U), so φ(z 0 ) U. and inductively φ 2 (z 0 ),,φ n (z 0 ) U,where φ n (z 0 ) = φ ( φ n 1 (z 0 )). The sequence z 0, φ(z 0 ),,φ n (z 0 ) is Cauchy, as follows from the contracting hypothesis. Hence we can set z = lim n φ n (z 0 ). By the continuity of φ, φ(z ) = z. The formula is easily by induction on n. Then d(z 0, φ n (z 0 )) < M(1 n) (4. 1) d(z 0, z ) = d(z 0, lim n φn (z 0 )) = lim n d(z 0, φ n (z 0 )) < M. (4. 2) Lemma 4.2 There exists η > 0, such that for any P C ω (Th d, U(n)) (h > 0) satisfies P I h < η, we can find F C ω (Th d, u(n)), such that P(.) = ef(.), and F h 2 P I h. (4. 3) Proof: For the map Φ : gl(n, C) GL(n, C), X e X, there exists open neighborhoods U gl(n, C) around 0 and V GL(n, C) around I, such that both Φ : U V and Φ U u(n) : U u(n) V U(n) are diffeomorphisms, the derivative for Φ satisfies max{ (X) I, 1 (X) I } < 1 2 (4. 4) for any X gl(n, C). Note 1 (X) is the derivative of Φ 1 at e X. Let D r = {X gl(n, C) : X < r}. There exists η > 0 such that D η U. Let D h,r = {F C ω h (Td, gl(n, C)) : F h < r} and for any fixed P C ω h (Td, GL(n, C)) define the map Ψ P as Then for any F, G D h,r Ψ P : D h,r C ω h (T d, gl(n, C)) (4. 5) F P + F e F (4. 6) Ψ P (F)(x) Ψ P (G)(x) (F(x) G(x)) (ξ x )(F(x) G(x)) (4. 7) 14

16 where ξ x {tf(x) + (1 t)g(x) : 0 t 1} D r. And thus Ψ P (F) Ψ P (G) h 1 2 F G h. (4. 8) Now d(0, C ω h (Td, gl(n, C)) D h,r ) r where d is the distance in C ω h (Td, gl(n, C). Assume r < η and d(0, Ψ P (0)) = d(0, P I) = P I h < 1 2 r, by Lemma 4.1, note Cω h (Td, gl(n, C)) is a complete metric space, there exist F D h,r such that Ψ P (F ) = F P = e F and the inequality followed. Moreover, if P C ω h (Td, U(n)), the F we obtained is also in C ω h (Td, u(n)). Acknowledgments. The authors would like to thank H.Eliasson for useful discussions and R.Krikorian for his interest of the result. References [El92] L.H.Eliasson: Floquet solutions for the one-dimensional quasiperiodic Schrödinger equation, Comm.Math.Phys., 146, ,1992 [El98] L.H.Eliasson: Reducibility and point spectrum for linear quasi-periodic skewproducts. In Proceedings of the International Congress of Mathematics, Vol.2 (Berlin,1998), number Extra Vol. 2, (electronic), 1998 [El01] L.H.Eliasson: Almost reducibility of linear quasi-periodic systems. In Smooth ergodic theory and its applications (Seattle, WA. 1999), Amer. Math. Soc., Providence, RI, 2001 [El02] L.H.Eliasson: Ergodic skew-systems on T d SO(3, R), Ergod. Th. Dynam. Sys., 22(5): , 2002 [HY04] H.He and J.You, Full Measure Reducibility for Generic One-Parameter Family of Quasi-Periodic Linear Systems, preprint, 2004 [JM82] R.Johnson and J.Moser: The rotation number for almost periodic potentials. Comm. Math. Phys. 84, no.3, , 1982 [Kr99b] R. Krikorian: Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans les groupes compacts. Ann. scient. Éc. Norm. Sup., 32: , 1999 [Kr99b] R. Krikorian: Astérisque, 259, 1999 [Kr01] R. Krikorian: Global density of reducible quasi-periodic cocycles on T SU(2), Annals of Mathematics (2)154 no 2, , 2001 [MP84] J.Moser and J.Poschel: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment.Math.Helv , 1984 [Sc69] J.Schwartz: Nonlinear functional analysis,