A PROOF OF THE INVARIANT TORUS THEOREM OF KOLMOGOROV

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1 A PROOF OF THE INVARIANT TORUS THEOREM OF KOLMOGOROV JACQUES FÉJOZ Abstract. A variant of Kolmogorov s initial proof is given, in terms of a group of symplectic transformations and of an elementary fixed point theorem. Let H be the space of Hamiltonians which are real analytic in neighborhoods of := {} in R n = {(θ, r} ( = R n /Z n. The vector field associated with H H is θ = r H, ṙ = θ H. For α R n, let K α be the subspace of Hamiltonians K H such that K is constant (i.e. is invariant and K = α: K α = {K H, c R, K(θ, r = c + α r + O(r 2 }, α r = α 1 r α n r n, where O(r 2 are terms of the second order in r, which depend on θ. Let also G be the space of symplectic transformations G real analytic in neighborhoods of in R n, of the following form: G(θ, r = (ϕ(θ, (r + ρ(θ ϕ (θ 1, where ϕ is an analytic transformation of fixing the origin (meant to straighten the flow on an invariant torus, and ρ is a closed 1-form on (or an irrotational vector field, meant to straighten an invariant torus. To be more precise, ρ(θdθ is a closed 1-form on. We fix α R n Diophantine ( < γ 1 and τ > n 1; see [6]: and k α γ k τ ( k Z n \ {}, k = k k n K o (θ, r = c o + α r + Q o (θ r 2 + O(r 3 K α such that the average of the quadratic form valued function Q o is non-degenerate: det Q o (θdθ. Theorem 1 (Kolmogorov [1, 4, 5]. For every H H close to K o, there exists (K, G K α G close to (K o,id such that H = K G in some neighborhood of G 1 (. See [3, 6, 9] and references therein for background. Here we present Kolmogorov s initial proof in its simplest form, but in the concise functional language of [2]. A beautiful paper claims to give the shortest complete KAM proof for perturbations of integrable vector fields available so far [7]. In fact, the only significant difference of this paper with Kolmogorov s induction is that at each step of the induction, Rüssmann [8] and Pöschel [7] optimize the error by taking a well chosen polynomial approximation of the right hand side of the cohomological equation; the convergence is slower and the range of convergence probably larger, but, as for the length of the proof, readers will judge by themselves. Define complex extensions C = Cn /Z n and C = Tn C Cn, and neighborhoods ( < s < 1 s = {θ C, max 1 j n Im θ j s} and s = {(θ, r C, max 1 j n max( Im θ j, r j s}. 1

2 2 JACQUES FÉJOZ For U = s or Tn s, we will denote by A(U the space of real holomorphic maps from the interior of U to C which extend continuously on U, endowed with the supremum norm s. Let H s = A( s, such that H = s H s. There exist s < s and ǫ > such that K o H s and, for all H H s with H K o s ǫ, (1 det 2 H (θ, dθ T r2 1 2 det 2 K o (θ, dθ n T r2. n Hereafter we assume that s is always s. Let K α s = {K H s K α, K K o s ǫ }; the vector space K s directing K α s identifies with R O(r 2. Let D s be the space of real holomorphic invertible transformations ϕ : s ϕ(tn s Tn C with ϕ( =, and Z s be the space of real holomorphic closed 1-forms on s (seen as maps s C n. Elements of G s = Z s D s define symplectic transformations of the phase space, (2 G : s C, (θ, r (ϕ(θ, (ρ(θ + r ϕ (θ 1, G = (ρ, ϕ, and of H s, K K G (the latter Hamiltonian being defined on G 1 ( s. Let d s := { ϕ A( s n, ϕ( = } with norm ϕ s := max θ s max 1 j n ϕ j (θ, be the space of vector fields on s which vanish at. Similarly, let ρ s = max θ s max 1 j n ρ j (θ on Z s. An element Ġ = ( ρ, ϕ of the sum g s = Z s d s (with norm ( ρ, ϕ s = max( ρ s, ϕ s identifies with the locally Hamiltonian vector field (3 Ġ : s C2n, (θ, r ( ϕ(θ, ρ(θ r ϕ (θ. Constants γ i, τ i, c i, t i below do not depend on s or σ. Lemma. If Ġ g s+σ and Ġ s+σ γ σ 2, then exp Ġ G s and exp Ġ id s c σ 1 Ġ s+σ. Proof. Let χ s A( s 2n, with norm v s = max θ s max 1 j 2n v j (θ. Let Ġ g s+σ with Ġ s+σ γ σ 2, γ := (36n 1. Using definition (3 and Cauchy s inequality, we see that if δ := σ/3, Ġ s+2δ = max ( ϕ s+2δ, ρ + r ϕ (θ s+2δ 2nδ 1 Ġ s+3δ δ/2. Let I s = [, 1] i[ s, s] and F := { f A(I s s 2n, (t, θ I s s, f(t, θ s δ }. By Cauchy s inequality, the Lipschitz constant of the Picard operator P : F F, f Pf, (Pf(t, θ = t Ġ(θ + f(s, θds is 1/2. Hence, P possesses a unique fixed point f F, such that f(1, = exp(ġ id and f(1, s Ġ s+δ c σ 1 Ġ s+σ, c = 6n. Also, exp Ġ G s because at all times the curve exp(tġ is tangent to G s (another proof uses the method of the variation of constants. Lemma 1 (Cohomological equation. For all (K, Ḣ Kα s+σ H s+σ, there exists a unique ( K, Ġ K s g s such that K + K Ġ = Ḣ and max( K s, Ġ s c 1 σ t1 (1 + K s+σ Ḣ s+σ.

3 A PROOF OF THE INVARIANT TORUS THEOREM OF KOLMOGOROV 3 Proof. We want to solve the linear (cohomological equation K + K Ġ = Ḣ. Write K(θ, r = c + α r + Q(θ r 2 + O(r 3 K(θ, r = ċ + K 2 (θ, r, ċ R, K 2 O(r 2 Ġ(θ, r = ( ϕ(θ, R + S (θ r ϕ (θ, ϕ χ s, Ṙ R n, Ṡ A( s. Expanding the equation in powers of r yields ( ( (4 ċ + (Ṙ + Ṡ α + r ϕ α + 2Q (Ṙ + Ṡ + K 2 = Ḣ =: Ḣ + Ḣ1 r + O(r 2, where the term O(r 2 on the right hand side does not depend on K 2. If g A( s+σ has zero average, there is a unique function f A(Tn s of zero average such that L α f := f α = g, and f s cσ t g s+σ, c = c γ,τ,n. Using the Diophantine condition and Cauchy s inequality to estimate Fourier coefficients, the unique formal solution indeed satisfies f s = g k eik θ ik α g s+σ k τ e k σ, γ s k Z n \{} and the wanted upper bound then follows from an elementary estimate [6]. Equation (4 is triangular in the unknowns and successively yields: Ṡ = L 1 α (Ḣ Ḣ(θdθ Ṙ = 1 2 ( T Q(θdθ 1 n T (Ḣ1 (θ 2Q(θ n Ṡ (θ dθ ϕ = ϕ 1 ϕ 1 (, ϕ 1 = L 1 α (Ḣ1 (θ 2Q(θ (Ṙ (θ + Ṡ ċ = T Ḣ(θdθ Ṙ α n K 2 = O(r 2. k The wanted estimate follows from Cauchy s inequality. Let us bound the discrepancy between the action of exp( Ġ and the infinitesimal action of Ġ. Lemma 2 (Quadratic error. For all (K, Ḣ Kα s+σ H s+σ such that (1 + K s+σ Ḣ s+σ γ 2 σ τ2, if ( K, Ġ K g s solves the equation K + K Ġ = Ḣ (lemma 1, then expġ G s, exp Ġ id s σ and (K + Ḣ exp( Ġ (K + K s c 2 σ t2 (1 + K s+σ 2 Ḣ 2 s+σ. Proof. Set δ = σ/2. Lemmas and 1 show that, under the hypotheses for some constant γ 2 and for τ 2 = t 1 + 1, we have Ġ s+δ γ δ 2 and exp Ġ id s δ. Let H = K + Ḣ. Taylor s formula says ( 1 H s H exp( Ġ = H H Ġ + (1 th exp( tġdt Ġ 2 or, using the fact that H = K + K + K Ġ, ( 1 H exp( Ġ (K + K = ( K + K Ġ Ġ + (1 th exp( tġdt Ġ2. The wanted estimate thus follows from the estimate of lemma 1 and Cauchy s inequality. End of the proof of theorem 1. Let B s,σ = {(K, Ḣ Kα s+α H s+σ, K s+σ ǫ, Ḣ s+σ (1 + ǫ 1 γ 2 σ τ2 } (recall (1.

4 4 JACQUES FÉJOZ By lemmas 1 and 2, the map φ : B s,σ K α s H s, φ(k, Ḣ = (K + K, (K + Ḣ exp( Ġ (K + K, K H satisfies, if ( ˆK, Ḣ = φ(k, Ḣ, ˆK K s c 3 σ t3 Ḣ s+σ, Ḣ s c 3 σ t3 Ḣ 2 s+σ. Proposition 3 in the appendix applies and shows that if H K o is small enough in H s+σ, the sequence (K j, Ḣj = φ j (K o, H K o, j, converges towards some (K, in K α s H s. K Ġ K Ḣ Ḣ H exp( Ġ K = K + K Let us keep track of the Ġj s solving with the K j s the successive linear equations K j +K j Ġj = Ḣ j (lemma 1. At the limit, K := K o + K + K 1 + = H exp( Ġ exp( Ġ1. Moreover, lemma 1 shows that Ġj sj+1 c 4 σ t4 j Ḣj sj, hence the transformations γ j := exp( Ġ exp( Ġj, which satisfy γ n id sn+1 Ġ s1 + + Ġn sn+1, form a Cauchy sequence and have a limit γ G s. At the expense of decreasing H K o s+σ, by the inverse function theorem, G := γ 1 exists in G s δ for some < δ < s, so that H = K G. Remark. The uniqueness property of lemma 1 and the estimate of lemma 2 show that if G is in some small neighborhood of the identity in G and K G K α then G = id. The local uniqueness of the pair (K, G such that H = K G follows directly. Appendix. Quadratic convergence Let (E s, s <s<1 and (F s, s <s<1 be two decreasing families of Banach spaces with increasing norms. On E s F s, set (x, y s = max( x s, y s. Fix C, γ, τ, c, t >. Let φ : B s,σ := {(x, y E s+σ F s+σ, x s+σ C, y s+σ γσ τ } E s F s be maps such that if (X, Y = φ(x, y, X x s cσ t y s+σ and Y s cσ t y 2 s+σ. In the proof of theorem 1, x s+σ C allows us to bound the determinant of Q(θdθ away from, while y s+σ γσ τ ensures that expġ is well defined. Lemma 3. Given s < s+σ and (x, y B s,σ such that y s+σ is small, the sequence (φ j (x, y j exists and converges towards a fixed point (ξ, in B s,. Proof. It is convenient to first assume that the sequence is defined and (x j, y j := F j (x, y B sj,σ j, for s j := s + 2 j σ and σ j := s j s j+1. We may assume c 2 t, so that d j := cσj t 1. By induction, and using the fact that 2 k = k2 k = 2, 2 j y j sj d j 1 y j 1 2 s j 1 y 2j d 2k+1 k ( y s+σ dk 2 k 1 = ( c(4σ t 2 j y s+σ. s+σ k j 1 Given that n µ2n 2µ if 2µ 1, by induction we see that if y s+σ is small enough, (x j, y j exists in B sj,σ j for all j, y j converges to in F s and the series x j = x + k j 1 (x k+1 x k converges absolutely towards some ξ E s with ξ s C. k

5 A PROOF OF THE INVARIANT TORUS THEOREM OF KOLMOGOROV 5 Thank you to the reviewer, Alain Albouy, Alain Chenciner, Ivar Ekeland, Andreas Knauf, Jessica Massetti and Éric Séré for their scrutiny and the improvements they have suggested. The author has been supported by the French ANR (projets KamFaible ANR-7-BLAN-361 and DynPDE ANR-1-BLAN-12. References [1] L. Chierchia. A. N. Kolmogorov s 1954 paper on nearly-integrable Hamiltonian systems. A comment on: On conservation of conditionally periodic motions for a small change in Hamilton s function. Regul. Chaotic Dyn., 13(2:13 139, 28. [2] J. Féjoz and M. Garay. Un théorème sur les actions de groupes de dimension infinie. C. R. Math. Acad. Sci. Paris, 348(7-8:427 43, 21. [3] J. Hubbard and Y. Ilyashenko. A proof of Kolmogorov s theorem. Discrete Contin. Dyn. Syst., 1(1-2: , 24. Partial differential equations and applications. [4] A. N. Kolmogorov. On the conservation of conditionally periodic motions for a small change in Hamilton s function. Dokl. Akad. Nauk SSSR (N.S., 98:527 53, [5] A. N. Kolmogorov. Preservation of conditionally periodic movements with small change in the Hamilton function. In Stochastic behavior in classical and quantum Hamiltonian systems (Volta Memorial Conf., Como, 1977, volume 93 of Lecture Notes in Phys., pages Springer, Berlin, [6] J. Pöschel. A lecture on the classical KAM theorem. In Smooth ergodic theory and its applications (Seattle, WA, 1999, volume 69 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, RI, 21. [7] J. Pöschel. KAM à la R. Regul. Chaotic Dyn., 16(1-2:17 23, 211. [8] H. Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete Contin. Dyn. Syst. Ser. S, 3(4: , 21. [9] M. B. Sevryuk. The classical KAM theory at the dawn of the twenty-first century. Mosc. Math. J., 3(3: , , 23. Dedicated to V. I. Arnold on the occasion of his 65th birthday. Université Paris-Dauphine, CEREMADE (UMR 7534 & Observatoire de Paris, IMCCE (UMR 828 address: jacques.fejoz@dauphine.fr