REDUCIBILITY OF THREE DIMENSIONAL SKEW SYMMETRIC SYSTEM WITH LIOUVILLEAN BASIC FREQUENCIES

. In this paper we consider the system ˙ x = ( A ( (cid:15) ) + (cid:15) m P ( t ; (cid:15) )) x,x ∈ R 3 , where (cid:15) is a small parameter, A,P are all 3 × 3 skew symmetric matrices, A is a constant matrix with eigenvalues ± i ¯ λ ( (cid:15) ) and 0, where ¯ λ ( (cid:15) ) = λ + a m 0 (cid:15) m 0 + O ( (cid:15) m 0 +1 )( m 0 < m ) , a m 0 (cid:54) = 0 , P is a quasi-periodic matrix with basic frequencies ω = (1 ,α ) with α being irrational. First, it is proved that for most of suﬃciently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that 0 ≤ β ( α ) < r, where β ( α ) measures how Liouvillean α is, r is the initial analytic radius, it is proved that for most of suﬃciently small parameters, this system can be reduced to constant system by means of a quasi-periodic change of variables.

In this paper, we are mainly concerned with the reducibility problems for analytic quasi-periodic skew symmetric system (1) with Liouvillean basic frequencies, i.e., ω = (1, α), where α is irrational, which means that the Diophantine condition (2) can be eliminated. When m 0 > 0(m > m 0 ), our non-degeneracy condition is more weaker. Moreover, the second Melnikov condition is not needed for the reducibility of three dimensional skew symmetric system, we only need the first Melnikov condition.
Let us first recall some well known results and development of reduction theory. Consider the systemẋ = A(t)x, x ∈ R n where A(t) is an n × n matrix which depends on time in a quasi-periodic way with basic frequencies ω = (ω 1 , . . . , ω s ). For s = 1, i.e., periodic case, the classical Floquet theory tells us that there exists a periodic change of variables such that the transformed system is a constant system.
For s > 1, i.e., quasi-periodic case, the system is not always reducible. The reducibility of quasi-periodic systems was initiated by Dinaburg and Sinai [7], who proved that the linear Schrödinger equation − y + q(ωt)y = Ey, y ∈ R, (4) or equivalently the following two-dimensional quasi-periodic system: is reducible for most of sufficiently large E, where the basic frequencies ω satisfy the Diophantine condition: where γ > 0, τ > s − 1 are constants. The reducibility of system (5) implies the existence of absolutely continuous spectrum of the Schrödinger operator Ly = − d 2 y dt 2 + q(ωt)y. Due to its importance in dynamical systems and the greatest relevance in the spectral theory of Schrödinger operator, the reducibility of quasi-periodic systems has been extensively investigated.
Liang-Xu [19] generalized the above results to high dimensional case. Johnson and Sell [15] proved that if the quasi-periodic coefficients matrix A(t) satisfies full spectrum condition, the system (3) is reducible. Jorba and Simó [16] considered the following linear differential equations: where A is a constant matrix with different nonzero eigenvalues λ 1 , . . . , λ n , P (t) is a quasi-periodic matrix with frequencies ω = (ω 1 , . . . , ω s ), and is a small parameter. They proved that if where γ > 0, τ > s − 1 are constants,λ i ( )(i = 1, . . . , n) are eigenvalues of A + P (t, ),P (t, ) is the average of P (t, ) with respect to t, then for most of sufficiently small parameters , the system (7) is reducible. In [17], Jorba and Simó extended the results of linear system to the nonlinear system, i.e., where A is a constant matrix with different nonzero eigenvalues, ) is a small perturbation with as a small parameter, h and f are quasi-periodic in t with frequencies ω = (ω 1 , . . . , ω s ). Later, Eliasson [9] proved that all quasi-periodic systems are almost reducible provided that the system satisfies Diophantine condition and is close to constant. Eliasson [8] obtained a full measure reducibility result for quasi-periodic Schrödinger equation. Krikorian [18] generalized the full measure reducibility result to linear systems with coefficients in Lie algebra of compact semi-simple Lie group. Her-You [13] and Chavaudret [5] established the full measure reducibility with coefficients in other groups. For the latest reducibility results of resonant cocycles and infinite dimensional quasi-periodic systems, we refer to [2,3,6,11] and the references therein.
During the development of reducibility of quasi-periodic systems and KAM theory, a lot of scholars are dedicated to weakening the non-degeneracy condition and non-resonant condition. Xu [28,29] obtained the reducibility of linear quasi-periodic system (7) in the case of multiple eigenvalues and more general non-degeneracy conditions, i.e., Moreover, the Diophantine condition (6) can also be weakened. Rüssmann [23] and Zhang-Liang [34] obtained the reducibility of Schrödinger equation under Brjuno-Rüssmann non-resonant condition: where γ > 0, and is continuous, increasing, unbounded function : [1, ∞) → [1, ∞) such that (1) = 1 and ∞ 1 ln (t) t 2 dt < ∞. is usually called Brjuno-Rüssmann approximation function. In particular, there have been some interesting aspects for 2-dimensional quasiperiodic systems. Without imposing any non-degeneracy condition, the reducibility of two dimensional quasi-periodic system was obtained in [26,30]. The idea is that if the non-degeneracy condition never occurs, then the non-resonant conditions are always the initial ones; if the non-degeneracy condition appears at some step, it can be kept on in the later steps. These particular phenomena [26,30] are the nature of 2-dimensional system, which don't hold in the high-dimensional case.
Moreover, the reducibility of quasi-periodic systems with Liouvillean frequencies has been obtained, i.e., ω = (1, α), where α ia irrational. A. Avila, B. Fayad and R. Krikorian [1] first introduced the method of CD bridge and proved the rotation reducibility of SL(2, R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. Zhou and Wang [36] used periodic approximation to study the reducibility of quasi-periodic GL(d, R) cocycles with Liouvillean frequencies, which can be viewed as a generalization of [1] to high dimension. Hou and You [14] considered a quasi-periodic linear differential system with two frequencies in sl(2, R) : and obtained almost reducibility and rotation reducibility of the above system, provided that the coefficients are analytic and close to constant. Furthermore, if the rotation number of the system and the basic frequencies ω = (1, α) satisfy Diophantine condition, the system is reducible. For other interesting results for 2-dimensional systems, we refer to [12,24] and the references therein. The proof of [14] is based on KAM method, the crucial observation is to analyze the structure of resonant terms, then to eliminate them by Floquet theory. Wang-You-Zhou [25] proved the existence of response solution for quasi-periodically forced harmonic oscillatorẍ with forcing frequencies ω = (1, α), where α is irrational. We [35] obtained the existence of n pair of conjugate quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies.
These particular phenomena make us realize that the 2-dimensional quasi-periodic system is very special. In this paper we will consider the reducibility for three dimensional skew symmetric system (1) with Liouvillean basic frequencies ω = (1, α), where α is irrational. As the matrix A has eigenvalues ±iλ( ) and 0, the system (1) looks like a two-dimensional system. But we don't know how to define the rotation number of three dimensional skew symmetric system, the proof method in [14] can't be adopted, so we consider the system (1) with small parameter.
The quasi-periodic systems in [1,14] are two dimensional, one can introduce the rotation number naturally and make good use of the property of rotation number. But for three dimensional skew symmetric system (1), it is difficult to define rotation number, which yields that there are essential obstructions in applying the methods in [1,14] to high dimensional situation. Comparing to [36], the former is discrete case, but this paper is continuous case. The method of periodic approximation for discrete case cann't be adopted to continuous case. Comparing to [25,35], our proof of this paper is reduced to the reducibility of a class of skew symmetric systems, so that the skew symmetric structure need to be preserved in every KAM steps. Moreover, the system (1) is a more general non-degeneracy quasi-periodic system, so KAM iteration is more complicated. Therefore, our non-degeneracy condition and non-resonant condition for the system are more optimized .
Before stating our results, we first give some notations and definitions. Usually, denote by Z and Z + the sets of integers and positive integers respectively. Denote by so(3, R) the set of 3 × 3 skew symmetric real matrices.
Furthermore, if F (θ) is analytic with respect to θ on D(r), we say that f (t) is analytic quasi-periodic on D(r). Denote by the average of f. Similarly, a function matrix P (t) = (P ij ) n×n is called analytic quasi-periodic on D(r), if all P ij (t) are analytic quasi-periodic on it. Denote by [P ] = ([P ij ]) n×n the average of P.
Remark 2. When β(α) = 0, the smallness of 0 doesn't depend on α, therefore, we not only weakens the Diophantine condition (2) to Liouvillean frequencies, but also improves the results in [10] to be non-perturbative in the case of two dimensional basic frequencies. In this sense, our theorem generalizes partial conclusions of [14] to high dimension.

2.
Proof of the main results.

2.1.
Outline of the proof. By Lemma 3.5 in the appendix, any skew symmetric matrix with eigenvalues ±iλ( ) and 0 is similar toλ( )J 1 , where the definition of J 1 is given in (10). Without loss of generality, suppose that A =λ( )J 1 . We divide the proof of Theorem 1.4 into two big steps. First, we prove that the system (1) can be reduced to a linear system with non-constant coefficients, which has a diagonal form: where The first big step can be achieved by infinite steps of KAM iteration. Second, we eliminate the non-resonant terms containing θ on the diagonal and transform the system (14) into a linear system with constant coefficients, This process can be obtained by only one step if 0 ≤ β(α) < r. Thus, it follows that the system (1) is reducible.

Homological equation.
Before giving the proof of Theorem 1.4, we first show how to use the method of diagonally dominant to solve the homological equation where W, P are all 3 × 3 skew symmetric matrices, B(θ), b(θ) have the same form as A, i.e., For τ > 2, we define and let (Q n ) be the selected subsequence of α in Lemma 3.2 with this given A. For r, γ > 0, we define where 0 <c, c 0 < 1 are constants, which will be defined later. and then the homological equation (15) has an approximate solution W (θ; ) with the estimate W r,Π ≤ 2C 3 (τ )γ −(Aτ +2) Q 6τ n+1 L − 1 240 P r,Π . Moreover, the error term P e = (P e,ij ) 3×3 with Proof. Let

Then by
For simplicity, denote Suppose that P has the following form: Therefore, the homological equation (19) can be written in the form of components: Note that the above homological equations have four cases. Let andP 13 (θ) = e iB(θ) P 13 (θ),P 32 (θ) = e iB(θ) P 32 (θ), W ij has a similar definition. Then the homological equation (19) is Instead of solving the equation (20), we first solve the approximation equation If we writeW and compare the Fourier coefficients of the equation (21), then for |k| < K, we have View (22) as a matrix equation where we use the matrix norm

2.3.
Elimination of non-resonant terms on the off-diagonal block. KAM step. Suppose that we are now in the n-th step, and in what follows the quantities without subscripts refer to those of the n-th step, while the quantities with subscripts "+" denote the corresponding ones of the (n+1)-th step. Thus we consider the following system dx dt where A =λ( )J 1 , B(θ) = Ξ(θ; )J 1 , For τ > 2, we define and let (Q n ) be the selected subsequence of α in Lemma 3.2 with this given A. For r, γ > 0, we define where [·] denotes its integer part, 0 <c < 1 is a constant which will be defined later, and c 0 =c 4 5 ·10 τ is a constant depending on τ,c.
We summarize one KAM step in the following lemma. The key point is to guarantee the non-resonant condition in KAM iteration by adjusting some parameters [16,17,19,28].
Proof. Before giving the proof, we first collect some useful estimates. Let and define inductively the following sequences E m =Ẽ 16 15 m−1 =Ẽ Once we have these parameters, there exists J = J(τ ), ε 0 = ε 0 (τ, r * ,c), and then we have the following useful estimates where C 3 (τ ) is the constant in Lemma 2.1.
In the following we first check the above estimates. For estimate (28) , and the choice of (Q n ) : Q n+1 ≤Q A 4 n , there exists ε 2 = ε 2 (τ, r * ,c) such that if ε 0 ≤ ε 2 , we havẽ For the first estimate of (29), if m ≥ m 0 , by the definition of σ m , the estimate holds apparently. If m < m 0 , by the definition of m 0 , K and σ m , we have For the second estimate of (29), if m ≥ m 0 , theñ
Then for any |k| < K, we can apply Lemma 2.1 to get an approximate solution W ν of (36) with the error term (T 1 KP ν ) e . That is to say, instead of solving (36), we first solve the approximation equation Moreover, the error term ( 16 15 ν 8 .
Below we prove B + (θ) has the same form as A. First an arbitrary skew symmetric matrix P can be written in the form Accordingly, define where γ + = γ/2, τ > 2τ + 2.

DONGFENG ZHANG, JUNXIANG XU AND XINDONG XU
For any given E > 0 satisfying (39), we define some sequences inductively: then for any ∈ Π n , the corresponding Diophantine condition holds for |k| < K n .
First, we claim that r n ≥ r * for all n. In fact, by our selection which implies that for any n ≥ 0, Second, by the choice of parameters we can verify that B n ,Q n0+n , P n satisfy that n0+n−1 ), βn P n rn,Πn ≤ βn M n = E n ≤ ε 0 γ J n L n . In the following we first check theses above estimates by induction. ByQ n+1 ≥ The definition L n and the estimate E 0 ≤ ε 0 ( γ 2 ) J yields that E n = L n L n−1 · · · L 1 E 0 ≤ L n 1 2 (n−1)J E 0 ≤ L n 1 2 (n−1)J ε 0 ( After setting the parameters, we will iterate KAM iteration infinitely and prove its convergence. For the first step, let B 0 = 0, P 0 = P, m 0 = m. By our selection Q n0 ≥ T ≥ T 0 γ − A 2 , and (39), we have Meanwhile, we can check (28) and (29) hold. Thus we apply Step Lemma 2.2, and get the transformation Φ 0 : Note that the above estimates means that all the conditions of Step Lemma 2.2 for the next step hold.
Inductively, there exists a subset Π n ⊂ Π such that for any ∈ Π n , the following Diophantine condition holds: and for any ∈ Π n there exists an orthogonal transformation Φ n = e Let Φ n = Φ 0 · · · Φ n−1 . Then, for any ∈ Π n , the orthogonal transformation x = Φ n x n reduces the system (1) into the following form: Measure estimates. In the following we give measure estimates of parameters. LetÂ By the above iteration, we have where γ n = γ 0 /2 n , τ > 2τ + 2.

2.4.
Elimination of non-resonant terms on the diagonal block. By infinite steps of KAM iteration, the system (1) can be reduced to a linear system with non-constant coefficients, which has a diagonal form: In this section, we will eliminate all the non-resonant terms containing θ on the diagonal by only one step. First, notice that β(α) has an equivalent definition (12), which implies the equation has an analytic solution if 0 ≤ β(α) < r * .
Then the above system becomes Because of the commutation e H(θ) and A, B * (θ), it follows that Moreover, for any ∈ Π * , we have This completes the proof of Theorem 1.4.

Appendix.
3.1. Continued fraction expansion. Let α ∈ (0, 1) be irrational. Define a 0 = 0, α 0 = α, and inductively for k ≥ 1, where [·] denotes its integral part, {·} denotes its fractional part. We define p 0 = 0, p 1 = 1, q 0 = 0, q 1 = a 1 , and inductively, The sequence (q n ) is the sequence of denominators of the best rational approximations for α, since it satisfies and where we use the norm 3.2. CD bridge. The motivation of this section is to introduce the concept of CD bridge, which first appeared in [1], and give some useful estimates. For detailed proofs we refer to [1,25]. For any α ∈ R \ Q, we fix a particular subsequence (q n k ) in the sequence of the denominators of α, which will be denoted by (Q k ) for simplicity. Denote the sequence (q n k +1 ) by (Q k ), and denote (p n k ) by (P k ).
Lemma 3.5. Let A = a 1 J 1 + a 2 J 2 + a 3 J 3 . Then the matrix A has three eigenvalues 0, ±ia, where a = a 2 1 + a 2 2 + a 2 3 . If x 3 ∈ R 3 is a unit eigenvector corresponding to eigenvalue zero, then there exist orthogonal unit vectors x 1 , x 2 ∈ R 3 , such that W = (x 1 , x 2 , x 3 ) is an orthogonal matrix, and Proof. First, the characteristic equation of A is det(λI − A) = λ[λ 2 + (a 2 1 + a 2 2 + a 2 3 )], so the eigenvalues are where a = a 2 1 + a 2 2 + a 2 3 . If a = 0, let W = I. Then If a = 0, note that zero is a simple eigenvalue, the vector satisfies that Ax 3 = 0, x 3 = 1, so the vector x 3 is a unit eigenvector corresponding to eigenvalue zero. Next we choose two orthogonal unit vectors x 1 , x 2 on the plane, which are both perpendicular to the vector x 3 , so that W = (x 1 , x 2 , x 3 ) is an orthogonal matrix, and Because orthogonal transformations preserve skew symmetric structure, i.e., they transform skew symmetric matrix into skew symmetric matrix, B is still a skew symmetric matrix and has the following form: Due to similar matrices have the same eigenvalues, we have m = a, i.e., Lemma 3.6. If Q is a skew symmetric matrix, then e Q is an orthogonal matrix.
Proof. According to the property of index matrix: if A, B can be exchanged, i.e., AB = BA, then e A · e B = e A+B . Therefore by e Q · (e Q ) T = e Q · e Q T = e Q+Q T = e Q−Q = I, e Q is an orthogonal matrix. Lemma 3.7. Consider the differentiable equationẋ = A(t)x, where A(t) is a skew symmetric matrix. Let x = Q(t)y, where Q(t) is an orthogonal matrix, which transforms the above differentiable equation intoẏ = By, then B is still a skew symmetric matrix.
Proof. Let x = Q(t)y. Then First note that Q is an orthogonal matrix, so Q −1 AQ is a skew symmetric matrix.
Second, differentiating the equation Q T Q = I with respect to t, we havė Q T Q + Q TQ = 0.
And because therefore, −Q −1Q is a skew symmetric matrix. Altogether, B is still a skew symmetric matrix.
Lemma 3.8. Suppose X satisfies the homological equation: where A, Y are both skew symmetric matrices, then X is a skew symmetric matrix.
Proof. Let X = k X k e i k,θ , Y = k Y k e i k,θ . Then comparing the Fourier coefficients of the above homological equation, we have Transposing on both sides of the equation gives Due to A, Y are all skew symmetric matrices, the equation (44) is equivalent to i.e., ∂ ω (−X T k ) + (−X T k )A − A(−X T k ) = Y k , k = 0.
By the equations (43) and (45), X k and −X T k are all the solutions of the equation According to the uniqueness of the solutions of equations, we have X k = −X T k , i.e., X k is a skew symmetric matrix. Therefore, X is a skew symmetric matrix. If Q ≤ 1/2, it follows that Therefore, (e Q ) −1 ≤ (e Q ) −1 − I + I ≤ 2.