REDUCIBILITY OF QUASI-PERIODICALLY FORCED CIRCLE FLOWS

. We develop a renormalization group approach to the problem of reducibility of quasi-periodically forced circle ﬂows. We apply the method to prove a reducibility theorem for such ﬂows.

1. Introduction. In this paper, we study the dynamics of quasi-periodically timedependent ordinary differential equations on the circle T 1 = R/(2πZ). These equations correspond to skew-product flows generated by vector fields on T d × T 1 whose dynamics are given byẋ = ω, where (x, y) ∈ T d × T 1 , ω ∈ R d and f : T d × T 1 → R is real analytic. An important problem in the dynamics of ordinary differential equations is to establish conditions under which one can analytically conjugate the flow φ t generated by (1) to the linear flow of a constant vector field. If that is possible, we say that f is analytically or C ω -reducible. An important conjugacy invariant is the rotation number of f , i.e., its time average along the orbit, for any (x, y) ∈ T d × T 1 (see Section 2. 3).
If f depends only on y (or ω = 0), the dynamical system given by (1) is integrable, since the second equation then yields an autonomous vector field on the circle. In that case, zeros of f correspond to the fixed points of the dynamics and all orbits converge to them. If f has no zeros, then all orbits are periodically winding around the circle and the flow is uniquely ergodic for some absolutely continuous invariant measure µ. The reducibility conjugacy can then be constructed using the solution φ t 2 of the second equation in (1), as (x, y) → (x, φ T 2π y 2 (0)), where T is the least period of the motion given by the frequency of the system 2πT −1 = rot f = T 1 f dµ. We are interested in the general case in which f depends on both x and y, i.e., non-autonomous circle flows. When d = 1, the system (1) corresponds to a periodic perturbation of the circle flow, i.e., to a flow on T 2 . Herman's theory [8] (further developed by Yoccoz [21]) implies that if rot f /ω satisfies Yoccoz's H arithmetical condition [21,22], then the system is C ω -reducible.
We will now restrict our considerations to ω ∈ R d \Q d , d > 1. In fact, we will focus only on incommensurate frequency vectors ω, i.e., vectors whose components are rationally independent. For incommensurate ω, the main difficulty in the analysis is related to the existence of small divisors. In this case, there are already some results obtained by KAM (Kolmogorov-Arnol'd-Moser)-type methods [1,2,16,19]. In this paper, we develop a different approach to the problem, based on a renormalization method. This renormalization approach is also different than that exploited in [5,6,7], which is based on resummation of a perturbation series in analogy to quantum field theory. While in KAM theory one typically encounters small divisors in a finite number of narrow regions surrounding some resonant planes, in the problem at hand, we encounter and develop an approach to deal with an infinite number of resonant planes (see Remark 2). The approach developed here should, therefore, also be useful for the construction of quasiperiodic solutions, i.e., invariant tori, for PDEs, where one necessarily has to deal with an infinite number of resonant planes.
On a space of vector fields X = (ω, f ) of the form (1), we define a renormalization operator R (see Section 3) as where η ∈ (0, 1) is the time rescaling parameter, U X is a change of variables chosen such that U * X (X), i.e., the pullback of X under U X , is in an appropriate normal form (see Section 3.2), and T is a scaling (x, y) → (T x, y), defined by a matrix T ∈ SL(d, R). The transformation R will be constructed such that it preserves the form of the vector field X = (ω, f ) and we will define the induced map R : . Note that f is a non-autonomous vector field on a circle and R defines a renormalization operator on a Banach space of these vector fields.
Since we will perform scaling with matrices T ∈ SL(d, R), we will consider functions with periodicity of a simple lattice Z in R d that is more general that 2πZ d . Functions that are invariant under Z-translations can be identified with functions on T d := R d /Z or, equivalently with quasiperiodic functions on R d with frequency module in the dual lattice V (the set of points v ∈ R d satisfying e iv·z = 1, for all z ∈ Z). For convenience, we will perform a linear change of coordinates in R d such that ω = (1, 0, . . . , 0). The lattice obtained from 2πZ d under this change of coordinates in R d will be denoted by Z 0 and its dual lattice by V 0 .
We consider vector fields X of the form X = (ω, f ) that are close to Y = (ω, θ), with f analytic on a complex neighborhood of D ρ,r of T d × T 1 characterized by | x i | < ρ and | y| < r. In the following, we will refer to these vector fields as vector fields of the form (ω, f ). In Section 2.2, we will introduce the spaces of vector fields of the form (ω, f ) with f analytic on D ρ,r , with frequency module in V, and the corresponding Banach spaces A ρ,r (V) of functions f . If r = ρ, we will denote these spaces and the corresponding domains simply by A ρ (V) and D ρ , respectively.
Let E be a projection operator onto the subspace of constant vector fields (either on the circle or on T d × T 1 ), given by the averaging of the function over T d × T 1 (see Section 2.2).
The main results of this paper can be summarized in the following theorem.
Theorem 1.1. Let , r > 0 and let ω ∈ BC and θ ∈ DC ω . There exist a sequence of matrices T n ∈ SL(d, R), a sequence of time rescaling parameters η n ∈ (0, 1), and a corresponding sequence of renormalization operators R n , n ∈ N, of the form (3), such that the corresponding operators R n are analytic from an open neighborhood D n−1 of θ n−1 , where θ n = η −1 n θ n−1 , θ 0 = θ, in A ρn−1,rn−1 (V n−1 ), to A ρn,rn (V n ), where V n = T n V n−1 . The set W of infinitely renormalizable vector fields f 0 in D 0 , characterized by the property that f n = R n (f n−1 ) belongs to D n , for every n ∈ N, is the graph of an analytic function W : (I − E)D 0 → ED 0 that satisfies W (0) = θ 0 and DW (0) = 0. In particular, if f ∈ D 0 and rot f = θ, then f ∈ W. If ρ > + δ, ρ > r + δ and δ > 0, then every vector field X = (ω, f ) with f ∈ W ∩ A ρ (V 0 ) is analytically reducible via an analytic conjugacy of the form Γ X = I + (0, ψ X ), where I is the identity map and ψ X ∈ A δ (V 0 ), that conjugates the flow of X and the flow of Y = (ω, θ).
An immediate corollary of Theorem 1.1 is the following. Corollary 1.2. For every ω ∈ BC and θ ∈ DC ω , there is an open ball B centered at θ in A ρ (V 0 ), such that every vector field X = (ω, f ), with f ∈ B and rot f = θ, is analytically reducible to Y = (ω, θ).

Remark 1.
Notice that the renormalization operators R n are well-defined on {ω}× D n , which are open in the space of vector fields of the form (ω, f ), and that {ω}×W is the stable manifold for this sequence of renormalization operators.
Remark 2. The renormalization approach developed here is similar to, but technically more involved than, the renormalization approach to the construction of invariant tori for Hamiltonian and other vector fields [10,11,12,14,17,18], reducibility of skew-product flows on T d × SL(2, R) [15] and construction of lower-dimensional tori for Hamiltonian flows [13]. The small divisors encountered in these problems are produced by frequencies ν ∈ Z d that lie in the "resonant" regions, outside of certain "non-resonant" cones, surrounding some resonant planes, perpendicular to ω. In the case of maximal-dimensional KAM tori, the small divisors are given by |ω · ν| and, thus, there is only one such a plane [10,11,12,14]. The renormalization transformations (see Section 3) then eliminate the non-resonant modes of a vector field and transform some of the remaining resonant modes into non-resonant. In the case of reducibility of skew-product flows on T d × SL(2, R), there is an additional resonant plane surrounded by frequencies corresponding to small divisors |ω ·ν −2ρ|, where ±iρ are the eigenvalues of a matrix in the Lie algebra sl(2, R) [15]. In the case of lower-dimensional tori [13], there are finitely many resonant planes corresponding to small divisors |ω · ν + Ω · V |, where Ω ∈ R D is vector of normal frequencies and 0 < |V | ≤ 2. In the problem at hand, we encounter and develop a renormalization approach to deal with an infinite number of resonant planes, corresponding to small divisors of the form |ω · ν + kθ|, for k ∈ Z.
The paper is organized as follows. In Section 2, we introduce the spaces of vector fields that we consider. In Section 3, we construct the renormalization scheme and prove the convergence of vector fields of the form (1), on the stable manifold of the renormalization operator, towards the constant vector field. In Section 4, we construct analytic conjugacy between the flows of a vector field X = (ω, f ) on the stable manifold, and a constant vector field Y = (ω, θ).

Preliminaries.
2.1. Skew-product vector fields and changes of coordinates. Recall that we are interested in skew-product vector fields on T d × T 1 , of the form with ω ∈ R d and f : T d × T 1 → R. We will refer to T d as the base and to T 1 as the fiber. The dynamics generated by X on the base is trivially given by x → x + ω t mod Z. We will consider real analytic diffeomorphisms H ∈ Diff ω (T d ×T 1 ) which preserve the space of skew-product vector fields and are of the type where (x, y) ∈ T d × T 1 and h ∈ C ω (T d × T 1 , T 1 ). We call them skew-product diffeomorphisms.
The action of H on X = (ω, f ) is given by the pull-back As the form of the vector field is preserved, we abuse the notation in order to write the pull-back as acting on the fiber component of the vector field The flow φ t generated by H * X is related to the flow φ t of X by The vector fields and skew-product diffeomorphisms considered are real-analytic and, thus, can be extended to a complex domain.

2.2.
Spaces and norms. We will use · and | · | to denote the ∞ and 1 norms, respectively, of a vector in R n or C n .
Let ρ, r > 0 and let In this paper, we consider functions with periodicity of Z × 2πZ, where Z is a lattice Z ⊂ R d . Recall that for a lattice Z ⊂ R d , the dual lattice is defined as We will denote by N , the lattice Z which is dual to 2πZ. The norm of a function f , analytic on D ρ,r , that can be expanded as is given by Given any K ∈ N, we denote by I K f the truncation of f corresponding to the modes with |k| ≤ K, i.e, We denote by I the identity operator acting as If = f , and by E the average of f , given by the action The Banach space of functions f , analytic on D ρ,r , for which the norm f ρ,r is finite will be denoted by A ρ,r (V). Similarly, A ρ,r (V) is the Banach spaces of functions f , analytic on D ρ,r , for which the norm is finite. Whenever there is no ambiguity, we avoid writing V explicitly.
We present several properties of the above norms, which will be used throughout the paper without an explicit reference to them.
Proof. The proof of these inequalities is straightforward and will be omitted. In the proof of the last inequality, one uses Cauchy's estimate for the derivative.

Rotation number. Define the (fibered) rotation number of
where φ t 2 = π 2 φ t stands for the last component of a lift φ t of the flow φ t to the universal cover R d+1 . Some versions of the following claim can be found in the literature. We include here a version relevant for our systems.
Proof. Notice that, for every x ∈ T d and every t ∈ R, the map y → φ t 2 (x, y) is an orientation-preserving diffeomorphism of the circle satisfying φ t Assume that the rotation number exists for some (x, y). Hence, for all t > 0. Taking the limit t → +∞, one obtains that rot f (x, y) = rot f (x) does not depend on y. Taking, e.g., y = 0, it remains to show that rot f (x) exists for all x and does not depend on x.
. We want to show that lim t→+∞ a t (x)/t exists and is independent of x.
We begin with some estimates. For s, s > 0 and (p, q) ∈ Z × (2πZ) satisfying We can easily bound the absolute values of the first two terms. For the first one, by noticing that whose absolute value is bounded from above by s times The second term is bounded by Taking the limit t → +∞, the first term on the right hand side approaches zero. Using Birkhoff's ergodic theorem, since the base flow x → x+ωt mod Z is uniquely ergodic with respect to the Lebesgue measure dm, we obtain Finally, taking s → +∞, this shows that the rotation number rot f (x) exists and that rot f (x) = T d rot f dm does not depend on x. We will use the following properties of the rotation number.
This proves the first part of the claim.
If we denote by φ t the flow of τ H * f , by identity (9), we have This completes the proof.

Arithmetics.
Lemma 2.4. If τ, κ > 0, there is κ > 0 such that for any C > 0 the Lebesgue measure of the complement of DC ω (C, τ, κ) is bounded from above by κC. Furthermore, Recall that ν = max i |ν i |. Clearly, ν ≤ |ν|. Moreover, for each n ∈ N, one has the following estimate on the cardinality Therefore, the Lebesgue measure of ν,k I ν,k is bounded by κC for some constant κ > 0, depending on d, τ and κ.
The measure of the above set approaches zero when C → 0. The second claim is now immediate.
3. Renormalization. In this section, we construct the renormalization scheme. We first define the non-resonant and resonant modes of a vector field and construct a change of coordinates that, via the pullback, eliminates non-resonant modes of a vector field. We then perform a scaling of the phase space that can produce some non-resonant modes of the transformed vector field. These two transformations, together with a time rescaling, form a one-step renormalization operator. Finally, we construct the stable manifold for a sequence of renormalizations operators.
3.1. Resonant cones. As explained in the introduction, we will perform a transformation of our coordinate system such that ω ∈ R d takes the form ω = (1, 0, . . . , 0). In this coordinate system, the lattice V 0 will be a lattice in R d which does not coincide with Z d .
At each renormalization step, we will perform the following phase space scaling and x = x + x ⊥ is the decomposition of x into component x parallel and x ⊥ perpendicular to ω. Notice that we will not scale the coordinate y, and all the functions that we consider will be periodic in y with period 2π; the lattice dual to 2πZ, the lattice N = Z, will be fixed throughout the paper.
Under the scaling, the lattice V is transforms into T V.
Definition 3.1. Given σ, K > 0 and a pair of lattices V and N , in R d and R, respectively, the nonresonant index set I − is defined as the set of pairs (v, k) ∈ (V, N ) such that |ω · v| > σ|v| and |k| ≤ K, or v = 0 and 0 < |k| ≤ K. The resonant index set I + is defined as the complement of I − in V × N .
Let I − be the projection operator onto the subspace spanned by modes (v, k) ∈ I − defined by the truncation The projection operator onto the subspace spanned by modes (v, k) ∈ I + is denoted by I + , and defined as I 3.2. Elimination of non-resonant modes. In this subsection, we construct a coordinate transformation U = U 1 such that U * f has no non-resonant modes. We construct this transformation using a homotopy method, which is different from the method used in [10,11,12,13,14,15]. Let ω ∈ R d , θ ∈ R, σ > 0, γ > 0, and there is an isotopy U t : D ρ,r → D ρ,r of real-analytic diffeomorphisms of the form U t (x, y) = (x, y + u t (x, y)) such that U 0 = I is the identity map, and where χ = 4t (ω, θ) Proof. Define the operator F : where U = I + u. The derivative of this operator is given by where D ω,θ = (ω, θ) · (∂ x , ∂ y ) and the dot denotes the dot product. We would like to determine a one-parameter family u t , with 0 ≤ t ≤ 1, satisfying F(u t ) = (1−t)F(u 0 ) and u 0 = 0.

SAŠA KOCIĆ AND JOÃO LOPES DIAS
Firstly, we will show that DF(0 Since the linear operator If r − r < 1, u ρ,r ≤ 1/2 and ∆ = 3 (ω, θ) Therefore, if Finally, by first differentiating F(u t ) = (1 − t)F(u 0 ) with respect to t and then integrating, we obtain whenever the family members u s satisfy the same smallness condition (22) as u above so that the derivative of F is invertible. Furthermore, u t ∈ I − A ρ,r is realanalytic for each t and satisfies Since and u t ∈ I − A ρ,r , by taking the I + projection of (23), and assuming f − θ ρ,r ≤ σ/(8γ), we obtain Since, by construction, we also have I − U * t f ρ,r = (1 − t) I − f ρ,r , and The second inequality in (19) follows. By taking the I − E projection of the identities (26) and (25), adding them up and using the fact that ( we similarly obtain the third inequality in (19).
By taking the E projection of identity (26), we obtain Taking into account that
3.3. Phase space and time rescaling. Consider the linear coordinate transformation T : (x, y) → (T x, y) rescaling the base torus, where T ∈ SL(d, R) is a matrix associated to ω, defined as in (14). In addition, we will perform a linear time rescaling t → η −1 t.
Since ω = η −1 T −1 ω, the joint action of T and time rescaling on X = (ω, f ) is given by We are interested in the action of this transformation on vector fields with no non-resonant modes, since the non-resonant modes are eliminated by a coordinate change constructed in Theorem 3.3.
Lemma 3.4. If 0 < ρ ≤ ηρ and 0 < η, β < 1, 0 < r ≤ r − σ/2, σ > 0, then T * defines a bounded linear operator from I + A ρ ,r (V) to A ρ ,r (T V), with the property that T * I Proof. Due to our choice of the norm, it suffices to verify the given bounds for a single modef v,k (x, y) = f v,k e ix·v+iyk labeled by (v, k). From the definitions of the scaling map it follows that In order to prove the first bound, assume that (v, k) belongs to I + and |k| ≤ K.

Renormalization transformations.
Following [12,13,15], we express the Brjuno condition on ω (and, thus, on V) in terms of the summability of the series of numbers for all positive integers n. Here κ and κ are two integer constants that will be specified later on. It follows from the definition that a n+1 /2 < a n < 2a n+1 , for all n ∈ N and, thus, a n+1 2 n+1 /4 < a n 2 n < a n+1 2 n+1 . In particular, a n 2 n is an increasing sequence.
We will define the scaling parameters as in [13], for all positive integers n. Since {a n } n∈N is a summable sequence of positive numbers, the sequence {A n } n∈N is well-defined, decreasing and converging to zero. We define, recursively, λ n = η n λ n−1 , with λ 0 = 1. These definitions imply the last bound in (16), since η n < β n < 1 for d > 1. These parameters are used to define the scaling maps T n and P n = T n · · · T 1 , at the n-th renormalization step, for each n ∈ N, as We also define T 0 = P 0 as the identity maps. Notice that the determinants |T n | = |P n | = 1, for all n ∈ N, by the choice of the scaling parameters. Given a lattice V 0 = V ⊂ R d , we define the lattice V n−1 = P n−1 V 0 , defining the frequency space of the functions that are going to be renormalized in the n-th step. The parameters L and used in the n-th renormalization step are and, thus, |ν | ≥ Ω n+κ by (28). Since we have Ω n+κ > e −an2 n+κ , this yields as claimed. Let C θ be a constant dependent of θ, that will be specified later on.
Definition 3.6. We define the resonant cone width parameter at the n-th renormalization step This definition immediately implies σ n > 0 and 2σ n L n−1 ≤ n−1 , for κ > C θ and all n ∈ N. Definition 3.7. Given the initial domain parameters , r > 0, we define the n-th step cut-off parameter The following proposition completes the verification of all bounds in (16).
Since {a n 2 n } n∈N is a growing sequence, the sequence {σ n } n∈N is decreasing. Notice also that for a fixed κ , and sufficiently large κ, we have 2 n+κ a n ≥ 2 n+κ (n + κ ) −2 ≥ c 2 κ n, for some constant c > 0 depending only on κ . This makes the sum ∞ n=1 σ n finite and, since by choosing κ sufficiently large, A 1 decreases, we can make this sum smaller than 1/2.
Remark 3. It follows from Proposition 3.8 and the fact that γ i ≥ 2 that r n > r/2.
Definition 3.11. We say that θ is DC ω (V) if there exist constants τ, κ > 0 and C > 0 such that for all v ∈ V\{0} and all k ∈ N .
Proof. In the first and the last inequality we have also used that µ n ≤ Cκ −N for any given C, N > 0, if κ is chosen sufficiently large.
From Theorem 3.3, it follows that there exists a universal constant R > 0 such that the n-th step renormalization operator R n is well-defined from an open ball B n−1 ⊂ A ρn−1,rn−1 (V n−1 ) around θ n−1 , of radius Rσ 2 n /(γ 2 n θ n−1 ), into A ρn,rn (V n ). We may choose the domain of the first step renormalization operator to be any ball D 0 ⊂ Rσ 2 1 /(γ 2 1 θ). Notice that the restriction of R n to EA ρn−1,rn−1 (V n−1 ) is a linear operator from EA ρn−1,rn−1 (V n−1 ) to EA ρn,rn (V n ) that will be denoted by L n .
The following claim follows directly from Theorem 3.3 and Lemma 3.4. In what follows, a domain D n−1 for R n is a subset of B n−1 described above, which is open in A ρn−1,rn−1 (V n−1 ) and contains θ n−1 . Given a domain D n−1 , for each R n , the domain D n−1 of for n ∈ N, is defined recursively as the subset of all functions in the domain of R n−1 that are mapped by R n−1 into the domain D n−1 of R n . By Theorem 3.16, these domains are open, non-empty, and the transformations R n are analytic.
To prove the following theorem, we apply the stable manifold theorem for sequences of mappings between Banach spaces that was proved in [11] (Section 6 therein).
Let D 0 be a domain whose existence is guaranteed by Theorem 3.17.
4. Analytic conjugacy. We say that a vector field where φ ω,θ is the linear flow of the constant vector field Y = (ω, θ). We refer to Γ X as the conjugacy between the flows of X and Y or the reducibility conjugacy for X. Consider a one-step renormalization operator R and a vector field X in the domain of R. If F is any map from We will restrict our consideration to maps F of the form F = I + ψ, where I is the identity map and ψ(x, y) = (0,ψ(x, y)). Notice, that M X preserves the form of these maps, since U X is also of the same form. Formally, if Γ R(X) is a conjugacy between the flows of R(X) and (ω, η −1 θ), then Γ X = M X (Γ R(X) ) is a conjugacy between the flows of X and the vector field (ω, θ). This can be seen easily from the identity Denote by A 0 (V) the Banach space of continuous functions F : D 0 → C d+1 , with frequency module in V, for which the norm F 0,V = v∈V,k∈N F v,k is finite, where F v,k are the Fourier coefficients of F .
Consider now a fixed but arbitrary vector field X = X 0 that belongs to the stable manifold W of our renormalization transformations, described by Theorem 3.17. Let X n = R n (X n−1 ), for n ≥ 1. In order to simplify the notation, we will write U k in place of the map U X k associated to the vector field X k and M k+1 in place of M X k . Our goal is to construct an appropriate sequence of functions Γ k ∈ A 0 (V k ), satisfying for all n ∈ N. Then, we will show that Γ 0 is the reducibility conjugacy for X 0 . Let us define B n , for every n ≥ 0, to be the vector space A 0 (V n ), equipped with the norm Denote by B n the unit ball in I + B n , centered at the identity I, and by B n /2 the ball of radius 1/2 in the same space.
where ψ = F − I. In order to estimate the norm of U n−1 − I, we can apply Theorem 3.3, with ρ = ρ n−1 and r = r n−1 , where r n ≤ r n−1 < r n−1 . By Theorem 3.17, for all n > 1, and for all f ∈ W ∩ B. Here, we have also used Proposition 3.14, and assumed that κ and then κ have been chosen sufficiently large, and that the neighborhood B of θ has been chosen sufficiently small (depending on κ and κ).
The first estimate in (56) and the final bound also hold for n = 1. The composition with I + g in Equation (55) can be controlled by Lemma 2.1, using the fact that g 0,Vn−1 ≤ η −1 n s n ψ n ≤ r n−1 /2, since ψ n ≤ 1, as we assume that F ∈ B n . Using s n /s n−1 = η n /4 , we obtain g n−1 ≤ η −1 n η n /4 ≤ 1/4 . From (56) we obtain U n−1 − I n−1 ≤ s −1 n−1 χ whereḡ = T n •ψ • T −1 n . Since g 0,Vn−1 ≤ ρ n−1 /2, using the Cauchy estimate on the derivative, we find Since r n−1 > r/2, we obtain a bound on the norm of this derivative analogous to that of (56). This, together with the fact that the inclusion map from B n into B n−1 is bounded in norm by η n /4, shows that and, consequently, DM n (F ) ≤ 1/3 , for all n ∈ N, and F ∈ B n . Below, we will make use of the following estimate on the difference between the flow for X = (ω, f ) and the flow for the constant vector field Y = (ω, θ). Proposition 4.2. Let τ > 0 and let X = (ω, f ) be a vector field with f ∈ A ,r (V), such that τ f − θ ,r < r < r. Then, for all t in the interval [−τ, τ ], Let φ n be the flow for the vector field X n . We start with the identity which follows from the relation (52) between the flow of a vector field and the flow of the renormalized vector field. , such that for every X = (ω, f ) with f ∈ W ∩ B, and for every n ≥ 1, the function φ s n • F • φ −s ω,θn belongs to B n , whenever F ∈ B n /2 and |s| ≤ χ −1/6 n . Proof. We will use the following easily verifiable identity Since, by assumption, ψ n ≤ 1/2, and ψ • φ −s ω,θn 0,Vn = ψ 0,Vn , we have ψ • φ −s ω,θn 0,Vn ≤ s n /2. By Proposition 4.2 and Theorem 3.17, we have the bound provided that the right hand side of this inequality is less than s n /2. This is certainly the case if f − θ ,r is chosen sufficiently small. The composition in (62) is well-defined since s n < r n . The third term on the right hand side of (62) can be bounded as which is smaller than 1/2, for any n ≥ 1, if f is sufficiently close to θ. The claim follows.
We will now construct the conjugacy.
Then, the limits Γ n = lim k→∞ Γ n,k exist in B n , are independent of the choice of F 0 , F 1 , . . ., and satisfy the identities (53). Furthermore, Γ 0 is the conjugacy between X and (ω, θ), and the map f → Γ 0 is analytic and bounded on W ∩ B.
Proof. By Proposition 4.1, the map M n : B n → B n−1 /2 contracts distances by a factor of at least 1/2. Thus, if 1 ≤ n < k < k , then the difference Γ n,k − Γ n,k is bounded in norm by 2 n−k+1 . This shows that the sequence k → Γ n,k converges in B n to a limit Γ n , which is independent of the choice of the functions F k . By choosing F k = Γ k for all k, we obtain the identities (53). The analyticity of f → Γ 0 follows, via the chain rule, from the analyticity of the maps used in our construction, and from uniform convergence. In order to prove that Γ 0 conjugates the flow of X to the linear flow of (ω, θ), we will use the identity (61). To be more precise, given a real number t, with |t| < 1, define t n = λ n t for all n ≥ 0. Proposition 4.3 allows us to iterate the identity (61), and get the identity for all k > 0. As proved above, the right (and thus left) hand side of this equation converges in A 0 (V) to Γ 0 . In addition, Γ 0,k → Γ 0 in A 0 (V), and the convergence is pointwise as well. Thus, since the flow φ t 0 is continuous, we have φ t 0 • Γ 0 • φ −t ω,θ = Γ 0 . This identity now extends to arbitrary t ∈ R, due to the group property of the flow, and the fact that composition with φ s ω,θ is an isometry on A 0 (V).
In what follows, the reducibility conjugacy Γ 0 for the vector field X = (ω, f ) with f ∈ W will be denoted by Γ X . For convenience, we extend the map f → Γ X to an open neighborhood of θ, by setting Γ X = Γ X , where X = (ω, f ) with f = (I + W )((I − E)f ).
Theorem 4.5. Let ρ > + δ, ρ > r + δ and δ > 0. Under the same assumptions as in Proposition 4.1, there exists an open neighborhood B of θ in A ρ (V 0 ), such that Γ X has an analytic continuation to Im x < δ and | Im y| < δ, for each X = (ω, f ) with f ∈ B. With this continuation, the map f → Γ X defines a bounded analytic map from B to A δ (V 0 ).
Proof. The proof of this theorem is analogous to the proof of Theorem 4.5 of [11]. Consider the translations R q,p (x, y) = (x + q, y + p), with q ∈ R d and p ∈ R. As before, for vector fields X = (ω, f ), R * q,p X denotes the pullback under R q,p ; the corresponding action on f is denoted by R * q,p X. For functions F : D 0 → D ρ , with ρ > 0, we define R * q,p F = R −1 q,p • F • R q,p . The renormalization operator R and the maps M X , defined in (51), satisfy Here, we have used the fact that the translations R * u are isometries on the spaces A r (V) and that the domain of R is translation invariant. This also implies that the manifold W is invariant under translations R * q,p , which is used in the second identity in (67).
As was explained above, we can extend the function f → Γ X to vector fields of the form X = (ω, f ) with f in an open neighborhood of θ, via Γ X = Γ X , where X = (ω, f ) and f = (I + W )((I − E)f ). If restricted to a sufficiently small open ball B ⊂ A ,r (V 0 ), centered at θ, the map f → Γ X is analytic and bounded on the whole B.
The construction of Γ 0 in the proof of Theorem 4.4, together with identities (67) and the invariance property W = W • R * q,p , shows that Γ R * q,p X = R * q,p Γ X , for all X = (ω, f ) with f ∈ B. Thus, if q ∈ R d and p ∈ R, then Γ X (q, p) = R q,p • Γ R * q,p X (0, 0), for f ∈ B. The idea now is to use the analyticity of map f → Γ X , to extend the right hand side of (68) to the complex domain Im q < δ and | Im p| < δ. Choose an open neighborhood B of θ in A ρ (V 0 ) such that R * q,p B ⊂ B for all (q, p) ∈ C d+1 of norm δ = ρ − or less. Then, the right hand side of (68), regarded as a function of (f, q, p), is analytic and bounded on the product of B with the strips q < δ and | p| < δ . Denoting this function by G, we have G(f, ·) ∈ A δ (V 0 ), for all f ∈ B . The analyticity of f → G(f, ·) is obtained now by using, for instance, a contour integral formula for (g(t) − g(0) − tg (0))/t 2 with g(t) = G(f + t f , ·).