MULTIDIMENSIONAL CONTINUED FRACTIONS, DYNAMICAL RENORMALIZATION AND KAM THEORY

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1 MULTIDIMENSIONAL CONTINUED FRACTIONS, DYNAMICAL RENORMALIZATION AND KAM THEORY KOSTYA KHANIN, JOÃO LOPES DIAS, AND JENS MARKLOF Abstract. The disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SLd, Z)\ SLd, R) the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. We explicitely construct renormalization schemes for a) the linearization of vector fields on tori of arbitrary dimension b) the construction of invariant tori for Hamiltonian systems. Contents 1. Introduction 2 2. Multidimensional continued fractions flows on homogeneous spaces Flows on homogeneous spaces A convenient parametrization Multidimensional continued fractions Siegel sets Dani s correspondence Diophantine conditions Norm estimates Hyperbolicity of the transfer matrices The resonance cone Renormalization of vector fields Definitions Resonance modes Basis change, time rescaling reparametrization Analyticity improvement Elimination of far from resonance modes Trivial limit of renormalization Analytic conjugacy to linear flow Renormalization of Hamiltonian flows Preliminaries Change of basis rescaling Far from resonance modes Analyticity improvement Elimination of far from resonance modes Convergence of renormalization Construction of the invariant torus 36 Appendix A. Elimination of modes 40 Date: 4/9/05. 1

2 2 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF A.1. Homotopy method for vector fields 40 A.2. Elimination of modes for Hamiltonians 44 Acknowledgements 50 References Introduction This paper has two main goals. The first one is to introduce a new multidimensional continued fraction algorithm that is ideally suited for different dynamical applications. The algorithm can be used in order to effectively deal with small divisors whenever quasi-periodicity with several frequencies is an essential feature of a problem. Our second goal is to demonstrate the strength of the algorithm by applying it to KAM theory. We use a renormalization approach to prove two theorems of KAM type. The method, being conceptually very simple, is also very general, allows us to consider a wide class of frequency vectors. For reasons of clarity we restrict our attention to vectors satisfying an explicit Diophantine condition valid for a set of vectors of full Lebesgue measure); generalizations to more general frequency vectors follow straightforwardly from the approach presented here, will be detailed in a separate publication. The classical continued fraction algorithm produces, for every irrational α R, a sequence of rational numbers p n /q n that approximate α up to an error of order 1/qn. 2 The first objective of this paper is to develop a multidimensional analogue that allows us to approximate any irrational α R d 1 by rational vectors. The drawback of all traditional multidimensional continued fraction algorithms is that it is not even known whether they provide simultaneous rational approximations for almost Lebesgue almost all α. In the case d 4 the only result in this direction is a recent computer-assisted proof of the almost everywhere strong convergence for ordered Jacobi-Perron algorithm [10, 11]. However, even in this case an explicit description of the set of bad vectors seems difficult. For example, the existence of noble vectors, that is vectors corresponding to a periodic continued-fraction expansion, for which approximations do not converge, is rather unsatisfactory. The algorithm we employ here does not suffer from such pathologies. Following Lagarias seminal ideas in [19], our approach is based on the dynamics of the geodesic 1 flow on the homogeneous space Γ\G with G = SLd, R) Γ = SLd, Z). Notice that Γ\G may be identified with the space of lattices in R d of covolume one or, equivalently, with the Teichmüller space of flat d-dimensional tori. The problem of multidimensional continued fractions may be formulated in the following way. Given a vector α R d 1 find a sequence of matrices T n) GLd, Z), n N, such that the cocycle corresponding to the products P n) = T n) T n 1)... T 1) exponentially contracts in the direction of the vector ω = α 1 ) R d exponentially exps in all other directions. Thus, the cocycle should have one negative Lyapunov exponent d 1 positive. In this spirit, our algorithm comprises the following steps: 1) With every α R d 1 associate the orbit {Ct) : t 0} SLd, R), where ) ) 1d 1 α 1d 1 e Ct) = t e d 1)t, 1.1) 1 The term geodesic is slightly inaccurate when d > 2. The orbits of the flow on Γ\G we are discussing here correspond in fact to geodesics on the unit cotangent bundle of the space Γ\G/ SOd) for a certain family of initial conditions. Only for d = 2 the cotangent bundle can be identified with Γ\G.

3 MULTIDIMENSIONAL RENORMALIZATION 3 1 d 1 denotes the d 1) d 1) unit matrix. 2) Fix a fundamental domain F of SLd, Z) in SLd, R). Given a sequence of times t 1 < t 2 <..., use classical reduction theory to find the matrices P n) SLd, Z) that map the points Ct n ) to F. 3) Define the nth continued fraction map by where α 0) = α α n 1) α n) = T n) 11 α n 1) + t n) 12, 1.2) t n) 21 α n 1) + t n) 22 T n) = T n) 11 t n) 12 t n) 21 t n) 22 ) SLd, Z) 1.3) is the nth transfer matrix defined by P n) = T n) P n 1). Since the action of SLd, R) on R d 1 by fractional linear transformation defines a group action, we have α n) = P n) 11 α + p n) 12, 1.4) p n) 21 α + p n) 22 where ) P n) = T n) T n 1) T 1) = P n) 11 p n) 12 p n) 21 p n) 22 SLd, Z). 1.5) Dani [5] Kleinbock-Margulis [15] observed that Diophantine properties of α translate to divergence properties of the corresponding orbit {ΓCt) : t 0} Γ\G in the cusps of Γ\G. We exploit these results to show that, under mild Diophantine conditions on α satisfied by a set of α of full Lebesgue measure, cf. Sec. 2.6), there is a sequence of times t n such that the transfer matrices T n) are uniformly hyperbolic in a sense made precise in Sec This fact allows us to develop renormalization schemes for vector fields Hamiltonian flows that had previously been constructed only in dimension one [21] or for very special choices of α [1, 16, 17, 20]. In particular, we obtain renormalization-based proofs of the following theorems. We say ω R d is Diophantine if there are constants ɛ > 0, C > 0 such that k d 1)1+ɛ) k ω > C, 1.6) for all k Z d {0}. Note that we may assume without loss of generality that ω is of the form ω = α 1 ) with α R d 1. Condition 1.6) then translates to a stard Diophantine condition on α, see Sec. 2.6) for details. Theorem 1.1. For any real analytic vector field v on T d, d 2, sufficiently close to a constant vector field with Diophantine vector ω R d, there is ɛ > 0, an analytic curve ɛ, ɛ) s p s R d an analytic conjugacy h isotopic to the identity between the flow generated by v + p s the linear flow φ t x) = x + t1 + s)ω on T d, t 0, for each s < ɛ. Moreover, the maps v h v p are analytic. Let us emphasize that the result holds for all analytic vector fields close to a constant one without any additional conditions, such as preservation of volume etc. The second theorem deals with the case of Hamiltonian vector fields. Let B R d, d 2, be an open set containing the origin, let H 0 be a real-analytic Hamiltonian function H 0 x, y) = ω y yqy, x, y) T d B, 1.7)

4 4 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF with ω R d a real symmetric d d matrix Q. It is said to be non-degenerate if det Q 0. Theorem 1.2. Suppose H 0 is non-degenerate ω is Diophantine. If H is a real analytic Hamiltonian on T d B sufficiently close to H 0, then the Hamiltonian flow of H leaves invariant a Lagrangian d-dimensional torus where it is analytically conjugated to the linear flow φ t x) = x + tω on T d, t 0. The conjugacy depends analytically on H. Outline of the proof of Theorem 1.1. Consider a vector field Xx, y) = ω + fx) + y where x T d, y R d is an auxiliary parameter. The vector field fx) is a sufficiently small analytic perturbation of a constant vector field. We may furthermore assume that ω = α 1 ) for some Diophantine α R d 1 ; this achieved by a rescaling of time. The aim is to find a value of parameter y = y ω such that the vector field Xx, y ω ) is linearizable to a constant vector field identically equal to ω by means of an analytic transformation of the coordinates on T d. Renormalization is an iterative process, we thus assume that after the n 1)th renormalization step the vector field is of the slightly more general form X n 1 x, y) = ω n 1) + f n 1 x, y) 1.8) where ω n 1) = αn 1) 1 ) α n 1) is given by the continued fraction algorithm, cf. 1.4). The Fourier modes of f n 1 are smaller than in the previous step, decay exponentially as k +. We define a cone of resonant modes by a relation I + n 1 = {k Z d : k ω n 1) σ n 1 k }. The nth step requires the following operations. 1) Eliminate all Fourier modes outside of the resonant cone I + n 1. 2) Apply a linear operator corresponding to a coordinate transformation given by the inverse transfer matrix T n) 1. 3) Rescale time to ensure that the frequency vector is of the form ω n) = αn) 1 ). The conjugate action on the Fourier modes is given by k T n) 1 k. It follows from the hyperbolicity of T n) that this transformation contracts for k I n 1 + if σ n 1 is small enough. This gives a significant improvement of the analyticity domain which results in the decrease of the estimates for the corresponding Fourier modes. As a result, all Fourier modes apart from the zero modes get smaller. To decrease the size of the latter, we choose a parameter y = y n in such a way that the corresponding zero modes vanish, then consider a neighbourhood of y-values centred at y n. That is, the auxiliary parameter y is used to eliminate an instability in the direction of constant vector fields. To get enough control on the parameter dependence we perform an affine rescaling of this parameter on every renormalization step. One can then show that the corresponding sequence of parameter domains is nested converges to a single point y = y ω for which the initial vector field is indeed linearizable. In order for the scheme to be effective, the sequence of stopping times t n the sizes of the resonant cones defined by the sequence of σ n must be chosen properly. Large intervals δt n = t n t n 1 improve hyperbolicity but, on the other h, worsen estimates for the norms T n), T n) 1. Similarly, if σ n is too small the elimination of non-resonant modes will give large contributions; on the other h, for large values of σ n the multiplication by T n+1) 1 will not yield a contraction for k I n +. As we shall show below a right choice of sequences {t n, σ n )} can be made, depending on the Diophantine properties of the vector ω.

5 MULTIDIMENSIONAL RENORMALIZATION 5 The idea of renormalization was introduced to the theory of dynamical systems by Feigenbaum [6] in the late 1970 s. In the case of Hamiltonian systems with two degrees of freedom MacKay proposed in the early 1980 s a renormalization scheme for the construction of KAM invariant tori [22]. The scheme was realized for the construction of invariant curves for two-dimensional conservative maps of the cylinder. An important feature of MacKay s approach is the analysis of both smooth KAM invariant curves so-called critical curves corresponding to critical values of a parameter above which invariant curves no longer exist. From the point of view of renormalization theory the KAM curves correspond to a trivial linear fixed point for the renormalization transformations, while critical curves give rise to very complicated fixed points with nontrivial critical behavior. MacKay s renormalization scheme was carried out only for a small class of Diophantine rotation numbers with periodic continued fraction expansion such as the golden mean). Khanin Sinai studied a different renormalization scheme for general Diophantine rotation numbers [14]. Both of the above early approaches were based on renormalization for maps or their generating functions. Essentially, the renormalization transformations are defined in the space of pairs of mappings which, being iterates of the same map, commute with each other. These commutativity conditions cause difficult technical problems, led MacKay [23] to propose the development of alternative renormalization schemes acting directly on vector fields. The same idea was realized by Koch [16] who proves a KAM type result for analytic perturbations of linear Hamiltonians H 0 x, y) = ω y, for frequencies ω which are eigenvectors of hyperbolic matrices in SLd, Z) with only one unstable direction. Notice that the set of such frequencies has zero Lebesgue measure in the case d = 2 corresponds to vectors with a quadratic irrational slope. Further improvements applications of Koch s techniques appeared in [1, 17, 20, 21, 7], emphasizing the connection between KAM renormalization theories. The results of this paper illustrate that such a programme can indeed be carried out in considerable generality. Another direction was followed in [18], presenting a computer-assisted proof of the existence of MacKay s golden mean critical renormalization fixed point in the context of Hamiltonian vector fields with two degrees of freedom. Other renormalization ideas have appeared in the context of the stability of invariant tori for nearly integrable Hamiltonian systems see e.g. [3, 8, 9] references therein). Inspired by quantum field theory an analogy with KAM theory, this approach uses an iterative resummation of Poincaré s Lindstedt series to prove its convergence. For the sake of transparency we have restricted our attention to Diophantine vectors ω. A more detailed analysis under weaker Diophantine conditions is, in principle, possible within the present framework. It is however a fundamental open problem to state a sharp i.e. the weakest possible) Diophantine condition under which the above conjugacy can be established. The answer to this question is known only in the classical case d = 2 [26] where the Diophantine condition is of Brjuno type. It would also be interesting to see whether the multidimensional continued fraction algorithm presented here will allow generalizations of other one-dimensional renormalization constructions. A concrete challenge is for instance the extension of the recent results on the reducibility of cocycles over irrational rotations by Ávila Krikorian [2]. In the next section we introduce the multidimensional continued fraction algorithm, include a discussion of its hyperbolicity properties required in the renormalization schemes. The remainder of the paper provides a detailed account of two exemplary

6 6 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF cases, the renormalization of vector fields Section 3) Hamiltonian flows Section 4). 2. Multidimensional continued fractions flows on homogeneous spaces 2.1. Flows on homogeneous spaces. Let us set G = SLd, R) Γ = SLd, Z), define the diagonal subgroup {E t : t R} in G, where with constants satisfying the conditions E t = diage r 1t,..., e r dt ) 2.1) r 1,..., r d 1 < 0 < r d, d r j = ) The right action of E t on the homogeneous space Γ\GΓ\G generates the flow j=1 Φ t : Γ\G Γ\G, ΓM ΓME t. 2.3) Since G is a simple Lie group with finite center, Φ t is ergodic mixing [24]. Let F G be a fundamental domain of the left action of Γ on G. Recall that, by definition of the fundamental domain of a free group action, P F = G, F P F = for all P Γ {1}, 2.4) P Γ hence, for any given M G, there is a unique family of P t) Γ such that holds for all t R. Mt) := P t)me t F 2.5) 2.2. A convenient parametrization. Let us consider those M G which can be written as ) ) 1 α A 0 M = ) β γ where A Mat d 1 R) the space of real d 1) d 1) matrices), α, β R d 1 are column vectors, γ R with γ > 0. This yields a local parametrization of G for the set { ) ) } 1 α A 0 G + := 0 1 G : A Mat β γ d 1 R), α, β R d 1, γ R >0, 2.7) which is particularly convenient for our purposes. All other matrices are either of the above form with γ < 0 instead, or may be written as ) ) 1 α A 0 M = S ) β γ where S Γ is a suitably chosen signed permutation matrix, i.e., every row every column contains one only one non-zero coefficient, which is either 1 or 1. In the following we will stay clear of the parameter singularity at γ = 0, thus may assume without loss of generality S = 1. To work out the action of a general element T G in the above parametrization, consider T : M M := T M 2.9) where ) T11 t T = 12, 2.10) t 21 t 22

7 M is as above MULTIDIMENSIONAL RENORMALIZATION 7 A short calculation yields the fractional linear action ) ) 1 α Ã 0 M =. 2.11) 0 1 β γ α α = T 11α + t 12 t 21 α + t 22, 2.12) γ γ = t 21 α + t 22 )γ, 2.13) more complicated expressions for Ã, β which will not be needed in the following Multidimensional continued fractions. Let t 0 = 0 < t 1 < t 2 <... be sequence of times, with gaps δt n := t n t n ) chosen large enough so that P t n ) P t n 1 ), where P t) is defined by 2.5). The sequence P n) := P t n ) of matrices in Γ may be viewed as the continued fraction approximants of the vector α, which are the best possible for suitable choices of a fundamental domain F times t n, see [19]. Let us furthermore put M n) := Mt n ) with Mt) as in 2.5), define α n), γ n) by the decomposition 2.6), i.e., by ) ) M n) 1 α n) A n) 0 = 0 1 β n) γ n). 2.15) From M n) = P n) ME tn 2.12), 2.13) we deduce where α n) = P n) 11 α + p n) 12, 2.16) p n) 21 α + p n) 22 γ n) = p n) 21 α + p n) 22 ) e r dt n γ 2.17) P n) = P n) 11 p n) 12 p n) 21 p n) 22 ). 2.18) It is evident that if the components of α, 1) are linearly independent over Q, then γ 0 implies γ n) 0 for all n 0. We shall later employ the transfer matrices T n) defined by P n) = T n) P n 1). Here, M n) = T n) M n 1) E δtn implies where α n) = T n) 11 α n 1) + t n) 12, 2.19) t n) 21 α n 1) + t n) 22 γ n) = t n) 21 α n 1) + t n) 22 ) e r dδt n γ n 1), 2.20) T n) = T n) 11 t n) 12 t n) 21 t n) 22 ). 2.21)

8 8 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF 2.4. Siegel sets. In dimensions d > 2 it is difficult to describe the geometry of a fundamental domain F. To overcome this problem, C. Siegel introduced simply connected sets S d G which have the property that they contain F are contained in a finite number of translates P F, P Γ. Consider the Iwasawa decomposition M = nak 2.22) where 1 u u 1d a n =... ud 1,d, a =.. 1 k SOd), with u ij, a j R, a j > 0, a 1 a d = 1. Then... a d 2.23) S d = {nak : n F N, a j 3 2 a j+1 > 0 j = 1,..., d 1), k SOd)} 2.24) is an example of a Siegel set [25]; here F N denotes a compact fundamental region of Γ N)\N, where N is the upper triangular group of elements of the form n as above Dani s correspondence. We assume from now on that r 1,..., r d 1 = 1, r d = d 1, i.e., E t = diage t,..., e t, e d 1)t ). 2.25) Let us denote by the maximum norm in R d 1. A vector α R d 1 is called badly approximable or of bounded type, if one of the following equivalent conditions is satisfied. i) There exists a constant C > 0 such that for all m Z d 1, k Z {0}. ii) There exists a constant C > 0 such that for all m Z d 1 {0}, k Z. kα + m d 1 k > C 2.26) m d 1 m α + k > C 2.27) The statements i) ii) are equivalent in view of Khintchine s transference principle [4] Chapter V). We recall Dani s correspondence in the following proposition cf. [5], Theorem 2.20). Proposition 2.1. The orbit {ΓME t : t 0}, with M as in 2.6), is bounded in Γ\G if only if the vector α is of bounded type. The reason why the parameters A, β, γ are irrelevant in the statement is that the family of matrices ) A 0 W t) = E t E β γ t 2.28) is bounded in G for all t 0. The boundedness of the orbit {ΓME t : t 0} implies of course that there is a compact set C G such that Mt) C for all t 0, with Mt) as in 2.5).

9 MULTIDIMENSIONAL RENORMALIZATION Diophantine conditions. A vector α R d 1 is called Diophantine, if there exist constants ɛ > 0, C > 0 such that m d 1)1+ɛ) m α + k > C 2.29) for all m Z d 1 {0}, k Z. It is well known that Diophantine vectors form a set of full Lebesgue measure [4]. Let us show that 2.29) implies the inequality k d 1)1+ɛ) k ω > C, 2.30) for all k Z d {0}, where ω = α 1 ), cf. 1.6). With k = m k ), 2.29) yields k ω > C m d 1)1+ɛ) C k d 1)1+ɛ) C k d 1)1+ɛ) 2.31) for all m Z d 1 {0}, k Z. In the case when m = 0, we have k 0 since k 0) thus 2.30) holds trivially. 2 Note also that 2.30) evidently implies 2.29), however with different choices for C in both inequalities. Following [15] we define the following function on G, δm) = inf km. 2.32) k Z d {0} It is easily checked that δm) is invariant under left action of Γ, may thus be viewed as a function on Γ\G. In terms of the Iwasawa parametrization 2.22) the Siegel set S d defined in 2.24) we have the following estimate. Lemma 2.2. For M = nak S d as in 2.22), 2.24), there are constants 0 < C 1 C 2 such that for all 0 < a d 1 C 1 a d δm) C 2 a d. 2.33) Proof. Since x x x for all x R d, we may prove the statement of the lemma for the function δ d M) = inf k Z d {0} km 2.34) instead. 3 Due to the rotational invariance of the Euclidean distance we may assume that k SOd) is the identity. Proof by induction. The statement trivially holds for d = 1. Therefore let us assume the assertion is true for dimension d 1. The jth coefficient of the vector km is j 1 km) j = k j + k i u ij )a j. 2.35) Since a 1 when a d 0, this implies that when taking the infimum in 2.32) we must take k 1 = 0 for all sufficiently small a d. Thus we now need to estimate j 1 ) j 1 inf max k j + k i u ij a j = a 1/d 1) 1 inf max k j + k i u ij )ã j k Z d 1 {0} 2 j d k Z d 1 {0} 2 j d 2.36) 2 Note that every admissible constant in 2.29) needs to satisfy C < 1/2; to see this, choose m = 1, 0,..., 0), k Z such that α 1 + k 1/2. 3 In the following, A B means there is a constant C > 0 such that A CB. If A B A we will also use the notation A B.

10 10 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF where k = k2,..., k d ), ã j = a 1/d 1) 1 a j so that ã 2 ã d = 1. Now j 1 inf max k j + k i u ij )ã j =: δ d 1 M) δ d 1 M) 2.37) k Z d 1 {0} 2 j d where M = ñã with 1 u u 2d ã ñ =... ud 1,d, ã = ã d. 2.38) It is easily checked that M S d 1, so by the induction hypothesis, for suitable constants 0 < C 1,d 1 C 2,d 1, we have C 1,d 1 ã d δ d 1 M) C 2,d 1 ã d, 2.39) provided ã d = a 1/d 1) 1 a d 1. So for a d sufficiently small a 1/d 1) 1 a d 1, we have C 1,d 1 a d δ d M) C 2,d 1 a d. 2.40) In the remaining case ã d > 1, all ã j are bounded from above below by positive constants, hence δ d 1 M) is bounded from above below by positive constants. Furthermore ã d > 1 implies a 1/d 1) 1 < a d,, in view of our choice of the Siegel set, a 1 1 = a 2 a d a d 1 d. So a d a 1/d 1) 1 < a d 2.41) the required bound follows from 2.36) also for the case ã d > 1. Lemma 2.3. Choose M as in 2.6) suppose α satisfies condition 2.29). Then there exists a constant C > 0 such that for all t 0 where δme t ) > C e θt 2.42) θ = d 1)ɛ d + d 1)ɛ. 2.43) Proof. Let us put k = m, k) with m Z d 1 k Z. Then δme t ) = inf me t, mα + k)e d 1)t) W t) m,k) Z d {0} inf me t, mα + k)e d 1)t ) 2.44) m,k) Z d {0} since W t), as defined in 2.28), is bounded in G for all t 0. sufficiently large Furthermore for t inf m,k) Z d {0} me t, mα + k)e d 1)t ) = inf m Z d 1 {0}, k Z me t, mα + k)e d 1)t ) 2.45) which, in view of the Diophantine condition 2.29), is bounded from below by inf m Z d 1 {0} me t, C m d 1)1+ɛ) e d 1)t ) = e θt inf m Z d 1 {0} x, C x d 1)1+ɛ) ) 2.46) where x = e θ 1)t m. We conclude the proof by noting that inf x, C x d 1)1+ɛ) ) inf y, C y d 1)1+ɛ) ) > ) m Z d 1 {0} y R d 1 {0}

11 MULTIDIMENSIONAL RENORMALIZATION 11 The fact that ɛ = 0 implies θ = 0 is consistent with Dani s correspondence. On the other h, θ < 1 for any ɛ < Norm estimates. Let denote the usual matrix norm Mx M := sup x 0 x. 2.48) Proposition 2.4. Choose M = M 0) as in 2.6), suppose α satisfies condition 2.29). Then there are constants c 1, c 2, c 3, c 4, c 5, c 6 > 0 such that for all n N {0} M n) c 1 exp[d 1)θt n ], 2.49) M n) 1 c 2 expθt n ), 2.50) P n) c 3 exp[d θ + 1 θ)t n ], 2.51) P n) 1 c 4 exp[d 1 + θ)t n ], 2.52) T n) c 5 exp[1 θ)δt n + d θ t n ], 2.53) T n) 1 c 6 exp[d 1)1 θ)δt n + d θ t n ]. 2.54) Proof. For any M S d as in 2.22) we have, for all 0 < a d 1, M a 1 = a 2 a d ) 1 a d 1) d, 2.55) M 1 a 1 d. 2.56) Combine this with Lemmas to obtain the bounds M n) 1 C 2 δm n) ) 1 = C 2 δm 0) E tn ) 1 < C 2 C 1 expθt n ) 2.57) M n) C d 1 2 δm n) ) d 1) = C d 1 2 δm 0) E tn ) d 1) < C d 1 2 C d 1) exp[d 1)θt n ]. 2.58) The remaining estimates follow immediately from 2.49), 2.50) the equations P n) = M n) E tn M 0) 1, T n) = M n) E δtn M n 1) ) Proposition 2.5. Choose M = M 0) as in 2.6), suppose α satisfies condition 2.29). Then there is a constant c 7 > 0 such that for all n N {0}, [ ] d 2 c 7 exp θ d 1) )t n γ n) c 1 exp[d 1)θt n ] 2.60) 1 θ with c 1 as in 2.49). Proof. The upper bound for γ n) follows from 2.49), since γ n) = M n) ) dd hence γ n) M n). From 2.17) the Diophantine condition 2.29) we have γ n) = γ exp[d 1)t n ] p n) 21 α + p n) 22 ) > C exp[d 1)t n ] p n) 21 d 1)1+ɛ). 2.61) Since 1 + ɛ = d 1 + θ 2.62) d 1)1 θ)

12 12 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF the proposition follows from the estimate 2.51) Hyperbolicity of the transfer matrices. Let p n) 21 P n) 2.63) ω n) = {ξ Rd : ξ ω n) = 0} 2.64) be the orthogonal complement of the vector ) ω n) α n) = R d. 2.65) 1 Lemma 2.6. For all ξ ω n 1), n N, T n) 1 ξ = exp δt n ) M n 1) M n) 1 ) ξ 2.66) Proof. This follows directly from the relation ) E δtn M n 1) A ξ = exp δt n ) n 1) ξ = exp δt 0 n ) M n 1) ξ 2.67) where ξ R d 1 comprises the first d 1 components of ξ. Proposition 2.7. Choose M = M 0) as in 2.6), suppose α satisfies condition 2.29). Then there is a constant Λ > 0 such that for all ξ ω n 1), n N, T n) 1 ξ 1 2 Λ exp ϕ n) ξ 2.68) with ϕ n = 1 θ)δt n d θ t n ) Proof. From Lemma 2.6, T n) 1 ξ exp δt n ) M n 1) M n) 1 ξ, 2.70) the proposition follows from the bounds 2.49), 2.50). Given any positive sequence ϕ 0, ϕ 1,..., the values t n that solve eq. 2.69) with t 0 = 0 are t n = 1 n 1 + β) n j ϕ j. 2.71) 1 θ where β = E.g., for constant ϕ n = ϕ > 0, we have { nϕ θ = 0) t n = ϕ [1 + dθ β)n 1] 0 < θ < 1). j=1 dθ 1 θ. 2.72) 2.73)

13 MULTIDIMENSIONAL RENORMALIZATION The resonance cone. As we shall see, a crucial step in our renormalization scheme is to eliminate all far-from-resonance modes in the Fourier series, i.e., all modes labeled by integer vectors outside the cone for a given σ n > 0. K n) = {ξ R d : ξ ω n) σ n ξ } 2.74) Lemma 2.8. Choose M = M 0) as in 2.6), suppose α satisfies condition 2.29). Then [ ] T n) 1 ξ Λ sup ξ K n 1) {0} ξ 2 + c 6σ n 1 e d δtn exp [ ] 1 θ)δt n + d θ t n 1, 2.75) for all n N. Proof. We write ξ = ξ 1 + ξ 2, where Firstly, ξ 1 = ξ ωn 1) ω n 1) 2 ωn 1), ξ 2 ω n 1). 2.76) T n) 1 ξ 1 T n) 1 ξ 1 = T n) 1 ξ ωn 1) ω n 1) σ n 1 T n) 1 ξ 2.77) since ξ K n 1) ω n 1) = α n 1), 1) 1. Hence in view of 2.54) T n) 1 ξ 1 c 6 σ n 1 exp[d 1)1 θ)δt n + d θ t n ] ξ. 2.78) Secondly, from Proposition 2.7 we infer T n) 1 ξ Λ exp [ 1 θ)δt n + d θ t n 1 ] ξ. 2.79) This proves 2.75). Remark 2.1. Note that if the t n are chosen as in 2.73), then for all ξ K n 1), n N ϕ > 0. σ n c 1 6 Λ exp d δt n ), 2.80) T n) 1 ξ Λ exp ϕ) ξ 2.81) 3. Renormalization of vector fields 3.1. Definitions. The transformation of a vector field X on a manifold M by a diffeomorphism ψ : M M is given by the pull-back of X under ψ: ψ X = Dψ) 1 X ψ. As the tangent bundle of the d-torus is trivial, T T d T d R d, we identify the set of vector fields on T d with the set of functions from T d to R d, that can be regarded as maps of R d by lifting to the universal cover. We will make use of the analyticity to extend to the complex domain, so we will deal with complex analytic functions. We will also be considering an extra variable related to a parameter.

14 14 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF Remark 3.1. We will be using maps between Banach spaces over C with a notion of analyticity stated as follows cf. e.g. [13]): a map F defined on a domain is analytic if it is locally bounded Gâteux differentiable. If it is analytic on a domain, it is continuous Fréchet differentiable. Moreover, we have a convergence theorem which is going to be used later on. Let {F k } be a sequence of functions analytic uniformly locally bounded on a domain D. If lim k + F k = F on D, then F is analytic on D. Let ρ, a, b > 0, r = a, b) consider the domain D ρ B r, 3.1) where D ρ = {x C d : Im x < ρ/2π} for the norm u = i u i on C d, { } d 1 B r = y = y 1,..., y d ) C d : y i < a y d < b. 3.2) Take complex analytic functions f : D ρ B r C d that are Z d -periodic on the first coordinate on the form of the Fourier series fx, y) = k Z d f k y)e 2πik x. 3.3) Its coefficients are analytic functions f k : B r C d with a continuous extension to the closure B r, endowed with the sup-norm: f k r = sup y B r f k y). The Banach spaces A ρ,r A ρ,r are the subspaces of such functions such that the respective norms f ρ,r = k Z d f k r e ρ k, f ρ,r = k Z d 1 + 2π k ) f k r e ρ k are finite. Also, write the constant Fourier mode of f A ρ,r through the projection Efy) = fx, y)dx = f 0 y) 3.4) T d into the projected space denoted by EA r. The norm of its derivative Df 0 is given by the operator norm Df 0 r = sup g r=1 Df 0 g r. Some of the properties of the above spaces are of easy verification. For instance, given any f, g A ρ,r we have: fx, y) f ρ,r f ρ,r where x, y) D ρ B r, f ρ δ,r f ρ,r with δ < ρ. In order to setup notations write, according to section 2, ω 0) = ω R d {0}, λ 0 = 1, for n N, ω n) = γ n) 1 M n) ) = λ n P n) ω = η n T n) ω n 1), 3.5) where λ n = γ γ n) ed 1)tn η n = λ n λ n )

15 MULTIDIMENSIONAL RENORMALIZATION 15 In the following, we will be interested in equilibria-free vector fields with a twist along the parameter direction. By rescaling this direction we will find the right parameter which guarantees the conjugacy to a linear flow. For a fixed n N {0}, we will be studying vector fields of the form where f A ρ,r Xx, y) = X 0 ny) + fx, y), x, y) D ρ B r, 3.7) X 0 ny) = ω n) + γ n) 1 M n) y. 3.8) We drop the second coordinate of the vector field because it will always be equal to zero there is no dynamics along the parameter direction.) The linear transformation on y deforms the set B r along the directions of the columns of M n) see 2.15)). In particular, its dth column corresponds to ω n). For the space of the above vector fields we use the same notation A ρ,r the same norm ρ,r without ambiguity Resonance modes. Given σ n > 0 we define the far from resonance Fourier modes f k as in 3.3) with respect to ω n) to be the ones whose indices k are in the cone I n = {k Z d : k ω n) > σ n k }. 3.9) Similarly, the resonant modes correspond to the cone I + n = Z d I n. 3.10) It is also useful to define the projections I + n I n on A ρ,r A ρ,r by restricting the Fourier modes to I + n I n, respectively. The identity operator is I = I + n + I n. Moreover, take A n = sup k I + n {0} T n+1) 1 k. 3.11) k A useful property of the above cones is included in the Lemma below. Lemma 3.2. If k In y B rn with r n = a n, b n ), ) 1 a n σ n 2 b n γ n) M n) 1 b n < 1 2, 3.12) then Proof. For every y B rn k I n, k X 0 n y) σ n > k. 3.13) 2 k ω n) + γ n) 1 M n) y) = 1 + y d )k ω n) + γ n) 1 k M n) y 1,..., y d 1, 0) Our choice of a n yields 3.13). > 1 b n )σ n k a n γ n) 1 M n) k. 3.14)

16 16 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF 3.3. Basis change, time rescaling reparametrization. The fundamental step of the renormalization is a transformation of the domain of definition of our vector fields. This is done by a linear change of basis coming essentially from the multidimensional continued fraction expansion of ω see section 2), a linear rescaling of time because the orbits take longer to cross the new torus, a change of variables for the parameter y in order to deal with the zero mode of the perturbation. Let ρ n 1, a n 1, b n 1 > 0, r n 1 = a n 1, b n 1 ) consider a vector field with f A ρn 1,r n 1. changes: Xx, y) = X 0 n 1y) + fx, y), x, y) D ρn 1 B rn 1, 3.15) We are interested in the following coordinate time linear x T n) 1 x, t η n t. 3.16) Notice that negative time rescalings are possible, meaning that we are inverting the direction of time. In addition to 3.16) we will use a transformation on y, a map y Φ n X)y) depending on X in a way to be defined later. Therefore, consider the transformation L n x, y) = T n) 1 x, Φ n X)y)), x, y) C 2d, 3.17) that determines a vector field in the new coordinates as the image of the map That is, for x, y) L 1 n D ρn 1 B rn 1, X L n X) = η n L nx. L n X)x, y) = η n T n) [ω n 1) + γ n 1) 1 M n 1) Φ n X)y) + f 0 Φ n X)y)] + η n T n) f f 0 ) L n x, y). 3.18) In order to eliminate the k = 0 mode of the perturbation of X in the new coordinates to normalise the linear term in y to γ n) 1 M n) y, using the definitions of T n) η n we choose ) 1 Φ n X): y Id +γ n 1) M n 1) 1 f 0 e dδt n y 1,..., e dδtn y d 1, y d ), 3.19) if possible. Hence, where L n X)x, y) = X 0 nx, y) + L n f f 0 )x, y), 3.20) L n : f η n T n) f L n. 3.21) Denote by µ the set of X A ρn 1,r n 1 such that f 0 rn 1 < µ. Lemma 3.3. Let r n = a n, b n ) µ n 1 > 0 such that ) ] a n e [a dδtn n γ n 1) M n 1) 1 µ n 1 ) b n b n γ n 1) M n 1) 1 µ n ) There exist an analytic map Φ n : µn 1 DiffB rn, C d ) such that, for each X µn 1, Φ n X) is given by 3.19) In case f 0 is real-analytic, Φ n X) R d is also real-valued. Φ n X)B rn ) B rn )

17 MULTIDIMENSIONAL RENORMALIZATION 17 Proof. For X A ρn 1,r n 1 with f 0 rn 1 < µ n 1 δ n 1 = δ n 1, δ n 1 ) with we have by the Cauchy estimate δ n 1 = µ n 1 γ n 1) M n 1) 1, 3.24) Df 0 rn 1 δ n 1 f 0 rn 1 δ n 1 < 1 γ n 1) M n 1) 1. So, F = Id +γ n 1) M n 1) 1 f 0 is a diffeomorphism on B rn 1 δ n 1. Now, if R 1 < a n 1 δ n 1 µ n 1, R 2 < b n 1 δ n 1 µ n 1 R = R 1, R 2 ), we have B R F B rn 1 δ n 1 ) F 1 B R ) B rn 1 δ n 1. Therefore, Φ n X) as given by 3.19) is a diffeomorphism on B rn by choosing R = e dδtn a n, b n ), thus we get 3.23). In addition, X Φ n X) is analytic from its dependence on f 0. When restricted to a real domain for a real-analytic f 0, Φ n X) is also real-analytic. Let the translation R z on C 2d be defined for z C d given by Notice that we have the following commutative relation: R z : x, y) x + z, y). 3.25) L nr z = R T n) z L n, z C d. 3.26) This also follows from the fact that Φ n is unchanged by the introduction of the translation R z Analyticity improvement. Lemma 3.4. If δ > 0 ρ n ρ n 1 A n 1 δ, 3.27) then L n as a map from I + n 1 E)A ρn 1,r n 1 µn 1 into I E)A ρ n,r n is continuous compact with L n η n T n) 1 + 2π ). 3.28) δ Remark 3.5. This result means that every vector field in I + n 1A ρn 1,r n 1 µn 1, i.e. a function on D ρn 1 B rn 1 into C d, has an analytic extension to T n) 1 D ρ n B rn 1. Proof. Let f I + n 1 E)A ρn 1,r n 1 µn 1. Then, ) f L n ρ n,r n 1 + 2π T n) 1 k f k M n rn e ρ n δ+δ) T n) 1k. 3.29) k I + n 1 {0} By using the relation ξe δ ξ δ 1 with ξ 0, 3.11) 3.23), we get f L n ρ n,rn 1 + 2π/δ) f k rn 1e An 1ρ I + n 1 {0} 1 + 2π/δ) f ρn 1,r n 1. n+δ) k 3.30) Finally, L n f ρ η n T n) f L n,rn n ρ,rn. n The above for D ρ n B rn is also valid for D ζ B rn, ζ > ρ n but satisfying a similar inequality to 3.27). Therefore, Ln = I J, where J : I + n 1 E)A ρn 1,r n 1 A ζ,r n is bounded as L n, the inclusion map I : A ζ,r n A ρ is compact. n,rn

18 18 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF For 0 < ρ n ρ n, consider the inclusion I n : A ρ n,rn A ρ n,rn 3.31) by restricting X A ρ n,r n to the smaller domain D ρ n B r n. When restricted to nonconstant modes, its norm can be estimated as follows. Lemma 3.6. If φ n 1 then Proof. For f I E)A ρ n,r n, we have I n f) ρ n,r n k 0 0 < ρ n ρ n logφ n ), 3.32) I n I E) φ 1 n. 3.33) 1 + 2π k ) f k rn e ρ n k φ k n φ 1 n f ρ n,r n. 3.34) 3.5. Elimination of far from resonance modes. The theorem below to be proven in Section A.1) states the existence of a nonlinear change of coordinates U, isotopic to the identity, that cancels the In modes of any X as in 3.7) with sufficiently small f. We are eliminating only the far from resonance modes, this way avoiding the complications usually related to small divisors. We remark that the parameter direction y is not affected by this change of coordinates. For given ρ n, r n, ε, ν > 0, denote by V ε the open ball in A ρ n+ν,r n centred at Xn 0 with radius ε. Theorem 3.7. Let r n be as in 3.12), σ n < ω n) ε n = σ { n ν 42 min 4π, For all X V εn there exists an isotopy σ n 72 ω n) U t : D ρn B rn D ρn+ν B rn, x, y) x + u t x, y), y), of analytic diffeomorphisms with u t in A ρ n,r n, t [0, 1], satisfying This defines the maps }. 3.35) 3.36) I n U t X = 1 t) I n X, U 0 = Id. 3.37) U t : V εn A ρ n,r n X Id +u t 3.38) U t : V εn I + A ρn,r n 1 t)i n A ρ n+ν,r n X U t X which are analytic, satisfy the inequalities U t X) Id ρ n,r n 42t σ n I n f ρn,r n U t X) X 0 n ρn,r n 3 t) f ρ n+ν,r n. If X is real-analytic, then U t X)R 2d ) R 2d. 3.39) 3.40)

19 MULTIDIMENSIONAL RENORMALIZATION 19 Remark 3.8. Further on we will be using the above result for t = 1. So that all far from resonance modes are eliminated. Recall the definition of the translation R z in 3.25). Lemma 3.9. In the conditions of Theorem 3.7, if x R d X V εn, then on D ρn,r n. U t X R x ) = R 1 x U t X) R x 3.41) Proof. Notice that R x D ρn B rn ) = D ρn B rn. If U t = U t X) is a solution of the homotopy equation 3.37) on D ρn B rn, then Ũt = Rx 1 U t X) R x solves the same equation for X = X R x, i.e. I X n Ũt = 1 t)i X, n on D ρn B rn Trivial limit of renormalization. Let a sequence of widths 0 < σ n < 1 of the resonance cones I n + be given. The nth step renormalization operator is thus R n = U n I n L n R n 1 R 0 = U 0, where U n is the full elimination of the modes in In as in Theorem 3.7 for t = 1). Notice that R n X 0 + v) = Xn, 0 for every v C d. From the previous sections the map R n on its domain is analytic. Also, in case a vector field X is real-analytic, the same is true for R n X). Fix the constants ν δ as in Theorem 3.7 Lemma 3.4, respectively, choose 0 < λ < 1. Take Θ n = min ε λ n σn 2 σ n γ n) σ n+1 γ n+1) n, n T i) 1, 2 λn M n) e dδt n+1 M n+1) 3.42) 1 + γ n) M n) 1 by assuming that the sequence of times t n guarantees that Θ n > 0. Now, write n B n = A i. 3.43) i=0 with A i given by 3.11). By recalling the inequalities 3.27) 3.32) we choose, for a given ρ 0 > 0, [ ] ρ n = 1 n 1 n 1 ρ 0 B i log φ i+1 ) δ + ν) B i, 3.44) B n 1 i=0 where { φ n = max 2 η n T n) 1 + 2πδ 1 ) Θ } n 1, ) Θ n is to be used in Lemma 3.6. Define the following function for every ω R d associated to the choice of σ n : Bω) = + i=0 B i log φ i+1 ) + δ + ν) + i=0 i=0 B i. 3.46) The convergence of the renormalization scheme now follows directly from our construction. Theorem Suppose that Bω) < ) ρ > Bω) + ν. There is K, b > 0 r n = a n, b n ) with a n > 0 b n > b > 0, such that if X is in a sufficiently small open ball around X 0 in A ρ,r0, then

20 20 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF i) X is in the domain of R n R n X) R n X 0 ) ρn,r n KΘ n X X 0 ρ,r0, n N {0}, 3.48) ii) for each s < b there exists in B rn 1 C d the limits p s nx) = lim Φ nr n 1 X))... Φ m R m 1 X))0,..., 0, s) 3.49) m + lim n + ps nx) 0,..., 0, s) = 0, 3.50) iii) the map X p n X) is analytic takes any real-analytic X into an analytic curve s p s nx) in R d. Proof. Let ξ > 0 ρ 0 = ρ ν ξ > 0 such that ρ 0 > Bω). Hence, by 3.44), we have R > 0 satisfying ρ n > RBn 1 1 for all n N. Denote by c the radius of an open ball in A ρ,r0 centred at X 0 containing X. If c ε 0 we can use Theorem 3.7 to obtain R 0 X) I + 0 A ρ0,r 0 with r 0 = a 0, b 0 ) satisfying 3.12) R 0 X) R 0 X 0 ) ρ0,r 0 2 X X 0 ρ+ξ,r 0 2ξ 1 X X 0 ρ,r0. Let K = 2ξΘ 0 ) 1 assume that c K 1 min{b 0 1 λ), 1 b 2 0}. So, 3.48) holds for n = 0. Now, with n N we choose the following r n : ) 1 γ n) a n = σ n 2 b n 1 0 b M n) n = b 0 ck λ i, 3.51) so that 1/2 > b n > b = b 0 ck1 λ) 1. The inequalities in 3.12) follow immediately. Moreover, 3.22) is also satisfied with µ n 1 = ckθ n 1 because a n 1 e dδtn a n 1 2 b 0)1 + γ n 1) M n 1) 1 )Θ n 1 i=0 1 + γ n 1) M n 1) 1 )ckθ n 1, b n 1 b n = ckλ n γ n 1) M n 1) 1 )ckθ n 1. Suppose that X n 1 = R n 1 X) I + n 1A ρn 1,r n 1 X n 1 X 0 n 1 ρn 1,r n 1 KΘ n 1 X X 0 ρ,r. 3.52) Since 3.22) holds, Lemmas are valid, together with 3.20) Lemma 3.6, can be used to estimate I n L n X n 1 ): I n L n X n 1 ) Xn 0 ρ n,r n η n T n) 1 + 2πδ 1 )φ 1 n KΘ n 1 X X 0 ρ,r0 = 1 2 KΘ n X X 0 ρ,r ) This vector field is inside the domain of U n as 3.12) 1c KΘ 2 n < ε n are satisfied. Thus 3.48) follows from 3.40). Denote by f n) 0 the constant mode of the perturbation term of X n. By Lemma 3.3, Φ n X n 1 ): B rn B rn 1 is given by y Id +g n ) diage dδtn,..., e dδtn, 1)y,

21 where MULTIDIMENSIONAL RENORMALIZATION 21 ) 1 g n = Id +γ n 1) M n 1) 1 f n 1) 0 Id 3.54) is defined on B r n with r n = e dδtn a n, b n ). So, for z B r n there is ξ B r n such that g n z) = [I + γ n 1) M n 1) 1 Df n 1) 0 ξ)] 1 [z γ n 1) M n 1) 1 f n 1) 0 0)] z = [I + γ n 1) M n 1) 1 Df n 1) 0 ξ)] 1 γ n 1) M n 1) 1 [Df n 1) 0 ξ) z + f n 1) 0 0)] 3.55) g n r n γ n 1) M n 1) 1 1 γ n 1) M n 1) 1 Df n 1) 0 r n The choice of r n means that min{a n 1 e dδtn a n, b n 1 b n } min By using 3.42) the Cauchy estimate, Thus, Df n 1) 0 r n f n 1) 0 rn 1 min{a n 1 e dδtn a n, b n 1 b n } r n Df n 1) 0 r n + f n 1) 0 r n ). 3.56) { } σn 1 γ n 1) M n 1), λn ) λ n 1 γ n 1) M n 1) ) g n r n λ n ) Writing y s = 0,..., 0, s), by induction we have m Φ n X n 1 )... Φ m X m 1 )y s ) = y s + diage dt i 1 t n 1 ),..., e dt i 1 t n 1 ), 1)g i ξ i ), i=n 3.60) for some ξ k B r k. Therefore, from 3.59), there exists p s nx) C d unless X is real which clearly gives p s nx) R d. In addition, p s nx) y s + i=n g i r i λn 1 1 λ. 3.61) The maps X p s nx) are analytic since the convergence is uniform. Lemma 3.3 gives us the nested sequence Φ n X n 1 )B rn ) B rn 1. So, as y s i N B ri, it follows that p s nx) B rn 1. Remark The above can be generalised for a small analyticity radius ρ by considering a sufficiently large N applying the above theorem to X = U N L N... U 1 L 1 U 0 X), where X is close enough to X 0. We recover the large strip case since ρ N is of the order of B 1 N 1. It remains to check that ρ N > Bω N) ) + ν. This follows from the fact that Bω N) ) = B 1 N 1 [Bω) B Nω)] where B N ω) is the sum of the first N terms of Bω) so that B N ω) Bω) as N +. Lemma If ω = α 1 ) in R d is diophantine, i.e. α satisfies 2.29) with exponent ɛ related to θ by 2.43) to β by 2.72)), then 3.47) is verified.

22 22 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF Proof. Let us set δt n = ξt n 1, σ n = exp cδt n ), n 1, where positive constants ξ, c will be chosen later. Obviously, t n = 1 + ξ)t n 1 = [1 + ξ)/ξ]δt n δt n = 1 + ξ)δt n 1. We shall assume that c < d1 + ξ), 3.62) so that σ n 1 expdδt n ) = exp cδt n 1 + dδt n ) = exp[d c/1 + ξ))δt n ] is much larger than Λ given by Proposition 2.7. Hence, using 2.75) we have [ A n 1 exp c 1 + ξ + d 1 θ) + dθ ξ ) δt n ]. 3.63) We next estimate ω n) ε n. It follows from 3.5) that ω n) M n) γ n) 1. Thus, using 2.49), 2.60) we have ) θ θ ω n) exp 1 θ d2 t n = exp 1 + ξ ) 1 θ d2 δt n. 3.64) ξ Since ω n) 1 one gets from 3.35) that ε n σn/ ω 2 n) which together with 3.64) implies [ exp 2c θ 1 + ξ ) ] 1 θ d2 δt n ε n exp 2cδt n ). 3.65) ξ Here X Y means that there exist two positive constants C 1, C 2 > 0 such that C 1 Y < X < C 2 Y. Using again 2.49), 2.60) we get [ σ n γ n) M n) exp c θ 1 + ξ ) ] 1 θ d2 δt n. 3.66) ξ Also, since M n+1) γ n+1), σ n+1 γ n+1) M n+1) expdδt n+1 ) exp[ c + d)1 + ξ)δt n]. 3.67) We shall assume that c ξ are chosen in such a way that c θ 1 + ξ 1 θ d2 ξ > c + d)1 + ξ), 3.68) so that σ n γ n) M n) σ n+1 γ n+1) M n+1) expdt n+1 ) σ n γ n) M n). 3.69) Inequality 3.68) is equivalent to the following condition c > θ 1 θ 1 + ξ ξ 2 d ξ d. 3.70) ξ Finally, we want A n to be small, hence, require the exponent in 3.63) to be negative c dθ + d 1 θ) + < ) 1 + ξ ξ Suppose that conditions 3.62), 3.70), 3.71) are satisfied. It follows immediately from the estimates above 3.6), 2.49), 2.50), 2.54), 2.60) that At the same time log Θ n, log Θ n 1, log T n), log η n δt n. 3.72) B n = n A i C n exp αt n+1 ), 3.73) i=0

23 where MULTIDIMENSIONAL RENORMALIZATION 23 α = c dθ d + 1 θ) 1 + ξ ξ > ) Since B n decays exponentially with t n log φ n grows at most linearly the series 3.47) converges. To finish the proof it is enough to show that conditions 3.62), 3.70), 3.71) can be satisfied. Indeed, since 0 < θ < 1 we can choose ξ so large that 1 θ dθ/ξ > 0 θ 1 θ 1 + ξ ξ 2 d ξ d < ) ξ It is easy to see that all three inequalities 3.62), 3.70), 3.71) are satisfied if 1+ξ)d β) < c < 1 + ξ)d, where β = 1 θ dθ/ξ > Analytic conjugacy to linear flow. As a consequence of Theorem 3.10, we obtain an analytic conjugacy between a vector field the linear flow, thus proving Theorem 1.1. In the following we always assume to be in the conditions of Section 3.6. Let r = r 0 = {X A ρ,r : X X 0 ρ,r < c} 3.76) inside the domain of R n for all n N {0}. By taking X, we denote X n = R n X) I + n A ρn,rn so that X n = λ n U 0 L 1 U 1 L n U n ) X), 3.77) where U k = U k I k L k X k 1 )) is given by Theorem 3.7 for t = 1 at the kth step L k is the linear rescaling as in 3.17) for X k 1. Denote by V n the coordinate change V n : x, y) P n) 1 x, Φ 1 X 0 )... Φ n X n 1 )y)) 3.78) set V 0 = Id. Thus, L n = V 1 n 1 V n X n = λ n V n U n ) V n 1 U n 1 V 1 n 1) V 1 U 1 V 1 1 ) U 0 X). 3.79) In particular, the y-coordinate is only transformed by the second component of V n. Notice that if X n = X 0 n for some n N, y = Φ 1 X 0 )... Φ n X n 1 )0,..., 0, s) C d, with s < b, corresponds to the parameter for which X is conjugated to 1 + s) ω n). The parameter value for the general case X n X 0 n 0 as n + is p s X) = p s 1X). Lemma There is an open ball B about X 0 in such that we can find a sequence R n > 0 satisfying R 1 = ρ, R n + 2π42KΘ 1/2 n X X 0 ρ,r R n 1 ρ n 1, X B, 3.80) P n 1) lim n + R 1 n Θn 1/2 = ) Proof. Let ρ = min ρ n. It is enough to check that Θn 1/2 λ n ρ n T i) 1 with 0 < λ < 1 taking R n = cλ n Θ 1/2 n for some positive constant c. This immediately implies 3.81) 3.80) by considering a small enough upper bound for H H 0 ρ,r.

24 24 K. KHANIN, J. LOPES DIAS, AND J. MARKLOF Let Diff per D ζ, C d ), ζ > 0, be the Banach space of Z d -periodic diffeomorphisms g : D ζ C d with finite norm g ζ = k g k e ζ k, where g k C d are the coefficients of the Fourier representation. It is simple to check that g P n) Rn g ρn. Denote by u n the analytic function u n : Diff per D ρn, C d ) X U n I n L n X n 1 )), p s n+1x)). 3.82) As p s n+1x) B rn, D ρn {y = p s n+1x)} is inside the domain D ρn,r n of U n I n L n X n 1 )) given in Theorem 3.7. Now, for each X, define the isotopic to the identity diffeomorphism W n X) = P n) 1 u n X) P n), 3.83) on P n) 1 D ρn. If X is real-analytic, then W n X)R d ) R d, since this property holds for u n X). We also have W n X 0 ) = Id. Lemma For all n N {0}, W n : B Diff per D Rn, C d ) is analytic satisfying W n X): D Rn D Rn 1 W n X) Id Rn 42KΘ 1/2 n X X 0 ρ,r, X B. 3.84) Proof. For any X, in view of 3.40) we get W n X) Id Rn = P n) 1 [u n X) Id] P n) Rn 42 σ n P n) 1 In L n X n 1 ) X 0 n ρn,r n. We can bound the above by 3.84). Now, for x D Rn X B, Im W n X)x) ImW n X)x) x) + Im x < W n X) Id Rn + R n /2π R n 1 /2π. So we have W n X): D Rn D Rn 1 W n X) Diff per D Rn, C d ). From the properties of U n, W n is analytic as a map from B into Diff per D Rn, C d ). Consider the analytic map H n : B Diff per D Rn, C d ) defined by the coordinate transformation H n X): D Rn D ρ0 as H n X) = W 0 X) W n X). 3.85) Lemma There exists c > 0 such that for X B n N, H n X) H n 1 X) Rn cθ 1/2 n X X 0 ρ,r. 3.86) Proof. For each k = 0,..., n 1, consider the transformations G k z, X) =W k X) Id) Id +G k+1 z, X)) + G k+1 z, X), G n z, X) =zw n X) Id), with z, X) {z C: z < 1 + d n } B, where we have c > 0 such that If the image of D Rn d n = c Θ 1/2 n X X 0 ρ,r 1 > 0. under Id +G k+1 z, X) is inside the domain of W k X), or simply G k+1 z, X) Rn R k R n )/2π,

25 MULTIDIMENSIONAL RENORMALIZATION 25 then G k is well-defined as an analytic map into Diff per D Rn, C d ), G k z, X) Rn W k X) Id Rk + G k+1 z, X) Rn. An inductive scheme shows that G n z, X) Rn R n 1 R n )/2π, By Cauchy s formula n 1 G k z, X) Rn W i X) Id Ri + z W n X) Id Rn i=k R k 1 R n )/2π. H n X) H n 1 X) Rn = G 0 1, X) G 0 0, X) Rn = 1 G 0 z, X) 2πi zz 1) dz z =1+d n/2 R n, H n X) H n 1 X) Rn 2 d n sup G 0 z, X) Rn z =1+d n/2 Θ 1/2 n X X 0 ρ,r. Consider CperR 1 d, C d ) to be the Banach space of the Z d -periodic C 1 functions between R d C d with norm f C 1 = max max D k fx). 3.87) k 1 x R d Lemma There exists C > 0, an open ball B B about X 0 an analytic map H : B Diff per R d, C d ) such that for X B, HX) = lim n + H n X) HX) Id C 1 C X X 0 ρ,r. 3.88) If X B is real-analytic, then HX) Diff per R d, R d ). Proof. As the domains D Rn are shrinking, we consider the restrictions of W n X) H n X) to R d, estimate their C 1 norms from the respective norms in Diff per D Rn, C d ). More precisely, for any X B, making use of Lemma 3.15, H n X) H n 1 X) C 1 max k 1 sup D k [H n X)x) H n 1 X)x)] x D Rn/2 4 R n H n X) H n 1 X) Rn, 3.89) which goes to zero by 3.81). Notice that here we have used Cauchy s estimate D 1 g ζ 2π/eδ) g ζ+δ with ζ, δ > 0. Therefore, it is shown the existence of the limit H n X) HX) as n +, in the Banach space CperR 1 d, C d ). Moreover, HX) Id C 1 X X 0 ρ,r. The convergence of H n is uniform in B so H is analytic. As the space of close to identity diffeomorphisms is closed for the C 1 norm, HX) is a diffeomorphism for any X sufficiently close to X 0, i.e. X B. The fact that, for real-analytic X, HX) takes real values for real arguments, follows from the same property of each W n X). To simplify notation, write π y X = X, y).