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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 324378, 18 pages doi:10.1155/2010/324378 Research Article Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings Dongfeng Zhang 1 and Rong Cheng 2 1 Department of Mathematics, Southeast University, Nanjing 210096, China 2 College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China Correspondence should be addressed to Dongfeng Zhang, zhdf@seu.edu.cn Received 19 April 2010; Revised 2 9 October 2010; Accepted 25 December 2010 Academic Editor: Roderick Melnik Copyright q 2010 D. Zhang and R. Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number μ>τ 2, where τ>m− 1 is the exponent in the small divisors condition and m is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves. 1. Introduction and Main Results In this paper we consider the following reversible mapping A: x 1 x h y f x, y , y 1 y g x, y , 1.1 where the rotation hy is real analytic and satisfies the weaker non-degeneracy condition h j 0 0, 0 <j<m, h m 0 / 0, 1.2 where fx, y and gx, y are real analytic and 2π periodic in x,thevariabley ranges in an open interval of the real line . We suppose that the mapping A is reversible with respect to the involution R : x, y → −x, y,thatis,ARA R. When some nonresonance and non- degeneracy conditions are satisfied and f, g are sufficiently small, the existence of invariant 2AdvancesinDifference Equations curve of reversible mapping 1.1 has been proved in 1–3. For related works, we refer the readers to 4– 6 and the references there. It is well known that reversible mappings have many similarities as Hamiltonian systems. Since many KAM theorems are proved for Hamiltonian systems, some math- ematicians turn to study the regular property of KAM tori with respect to parameters. One of the earliest results is due to P ¨ oschel 7, who proved that the KAM tori of nearly integrable analytic Hamiltonian systems form a Cantor family depending on parameters only in C ∞ -way. Because the notorious small divisors can result in loss of smoothness with respect to parameters involving in small divisors in KAM steps, we can only expect Gevrey smoothness of KAM tori even for analytic systems. Gevrey smoothness is a notion intermediate between C ∞ -smoothness and analyticity see definition below. Popov 8 obtained Gevrey smoothness of invariant tori for analytic Hamiltonian systems. In 9, Wagener used the inverse approximation lemma to prove a more general conclusion. Recently, the preceding result has been generalized to R ¨ ussmann’s non-degeneracy condition 10–12. Gevrey smoothness of the family of KAM tori is important for constructing Gevrey normal form near KAM tori, which can lead to the effective stability 8, 13. For reversible mappings, if h y / 0, the existence of a C ∞ -family of invariant curves has been proved in 1, 2. But in the case of weaker non-degeneracy condition 1.2,thereis no result about Gevrey smoothness. In this paper, we are concerned with Gevrey smoothness of invariant curve of reversible mapping 1.1. T he Gevrey smoothness is expressed by Gevrey index. In the following, we specifically obtain the Gevrey index of invariant curve which is related to smoothness of reversible mapping 1.1 and the exponent of the small divisors condition. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves. As in 7, 14, 15, we introduce some parameters, so that the existence of invariant curve of reversible mapping 1.1 can be reduced to that of a family of reversible mappings with some parameters. We write y p z, and expand hy around p,sothathyhp 1 0 h y t zdt,wherey t p tz,0≤ t ≤ 1, z varies in a neighborhood of origin of the real line .Weputωphp, fx, z; p 1 0 h y t zdt fx, p z, gx, z; pgx, p z and obtain the family of reversible mappings x 1 x ω p f x, z; p , z 1 z g x, z; p . 1.3 Now, we turn to consider this family of reversible mappings with parameters p ∈ Π,where Π ⊂ is a bounded interval. Before stating our theorem, we first give some definitions and notations. Usually, denote by and the set of integers and positive integers. Definition 1 .1. Let D be a domain of n .AfunctionF : D → is said to belong to the Gevr ey- class G μ D of index μμ ≥ 1 if F is C ∞ D-smooth and there exists a constant M such that for all p ∈ D, ∂ β p F p ≤ cM |β|1 β! μ , 1.4 where |β| β 1 ··· β n and β! β 1 ! ···β n !forβ β 1 , ,β n ∈ n . Advances in Difference Equations 3 Remark 1.2. By definition, it is easy to see that the Gevrey-smooth functions class G 1 coincides with the class of analytic functions. Moreover, we have G 1 ⊂ G μ 1 ⊂ G μ 2 ⊂ C ∞ , 1.5 for 1 <μ 1 <μ 2 < ∞. In this paper, we will prove Gevrey smoothness of function in a closed set, so we give the following definition. Definition 1.3. AfunctionF is Gevrey of index μ on a compact set Π ∗ if it can be extended as a Gevrey function of the same index in a neighborhood of Π ∗ . Define D s, r { x, z ∈ /2π × | | Im x | ≤ s, | z | ≤ r } , 1.6 and denote a complex neighborhood of Π by Π h p ∈ | dist p, Π ≤ h . 1.7 Now the function fx, z; p is real analytic on Ds, r × Π h .Weexpandfx, z; p as Fourier series with respect to x f x, z; p k∈ f k z; p e ikx , 1.8 then define f Ds,r×Π h k∈ f k r,h e s|k| , 1.9 where f k r,h sup | z | ≤r,p∈Π h f k z; p . 1.10 We write Fx, z; p ∈ G 1,μ Ds, r × Π ∗ if Fx, z; p is analytic with respect to x, z on Ds, r and G μ -smooth in p on Π ∗ . Denote T max p∈Π h |ω p|. Fix δ ∈ 0, 1 and τ>m− 1, and let μ τ 2 δ and σ 2/3 δ/τ1δ .LetW 0 diagρ −1 0 ,r −1 0 . 4AdvancesinDifference Equations Theorem 1.4. We consider the mapping A defined in 1.3, which is reversible with respect to the involution R : x, z → −x, z,thatis,ARA R. Suppose that ωp satisfies the non-degeneracy condition: ω j 00, 0 <j<m, ω m 0 / 0. Suppose that fx, z; p and gx, z; p are real analytic on Ds, r × Π h . Then, there exists γ>0 such that for any 0 <α<1,if f Ds,r×Π h 1 r g Ds,r×Π h ≤ γαs τ2 , 1.11 there is a nonempty Cantor set Π ∗ ⊂ Π, and a family of transformations V ∗ ·, ·; p : Ds/2,r/2 → Ds, r, ∀p ∈ Π ∗ , x ξ p ∗ ξ; p , z η q ∗ ξ, η; p , 1.12 satisfying V ∗ x, z; p ∈ G 1,μ Ds/2,r/2 × Π ∗ , and for any β ∈ , W 0 ∂ β p V ∗ − id Ds/2,r/2×Π ∗ ≤ cM β β! τ2δ γ 1/2 , 1.13 where M 2 τ2δ T 1τ 1 δ τ1δ /πα, the constant c depends on n, τ,andδ. Under these transformations, the mapping 1.3 is transformed to ξ 1 ξ ω ∗ p f ∗ ξ, η; p , η 1 η g ∗ ξ, η; p , 1.14 where f ∗ Oη, g ∗ Oη 2 at η 0.Thus,foranyp ∈ Π ∗ , the mapping 1.3 has an invariant curve Γ such that the induced mapping on this curve is the translation ξ 1 ξ ω ∗ p, whose frequency ω ∗ p satisfies that ∂ β p ω ∗ p − ω p ≤ cαM β β! τ2δ γ 1/2 s τ2 , ∀β ∈ , 1.15 kω ∗ p 2π − l ≥ α 2 | k | τ , ∀ k, l ∈ × \ { 0, 0 } . 1.16 Moreover, one has meas Π \Π ∗ ≤ cα 1/m . Remark 1.5. From Theorem 1.4, we can see that for any μ>τ 2, if is sufficiently small, the family of invariant curves is G μ -smoothintheparameters.TheGevreyindexμ τ 2 δ should be optimal. Remark 1.6. The derivatives in 1.13 and 1.15 should be understood in the sense of Whitney 16. In fact, the estimates 1.13 and 1.15 also hold in a neighborhood of Π ∗ with the same Gevrey index. Advances in Difference Equations 5 2. Proof of the Main Results In this section, we will prove our Theorem 1.4. But in the case of weaker non-degeneracy condition, the previous methods in 1, 2 are not valid and the difficulty is how to control the parameters in small divisors. We use an improved KAM iteration carrying some parameters to obtain the existence and Gevrey regularity of invariant curves of analytic reversible mappings. This method is outlined in the paper 7 by P ¨ oschel and adapted to Gevrey classes in 13 by Popov. We also extend the method of Liu 1, 2. KAM step The KAM step can be summarized in the following lemma. Lemma 2.1. Consider the following real analytic mapping A: x 1 x ω p f x, z; p , z 1 z g x, z; p , 2.1 on Ds, r × Π h . Suppose the mapping is reversible with respect to the involution R : x, z → −x, z,thatis,ARA R.Let0 <E<1, 0 <ρ<s/5,andK>0 such that e −Kρ E. Suppose ∀p ∈ Π, the following small divisors condition holds: kω p 2π − l ≥ α | k | τ , ∀ k, l ∈ × \ { 0, 0 } , 0 < | k | ≤ K. 2.2 Let max p∈Π h ω p ≤ T, h πα TK τ1 . 2.3 Suppose that f s,r;h 1 r g s,r;h ≤ αρ τ2 E, 2.4 where the norm · s,r;h indicates · Ds,r×Π h for simplicity. Then, for any p ∈ Π h ,thereexistsa transformation U: x ξ u ξ; p , z η v ξ, η; p , 2.5 which is affine in η, such that the mapping A is transformed to A U −1 AU: ξ 1 ξ ω p f ξ, η; p , η 1 η g ξ, η; p , 2.6 6AdvancesinDifference Equations where the new perturbation satisfies f s ,r ;h 1 r g s ,r ;h ≤ α ρ τ2 E , 2.7 with s s − 5ρ, ρ σρ, μ √ E, r μr, E cE 3/2 , α 2 ≤ α ≤ α, 2.8 where σ is defined in Theorem 1.4. Moreover, one has ω p − ω p ≤ , ∀p ∈ Π h . 2.9 Let α α − /2πK τ1 , and denote R k p ∈ Π | kω p 2π − l < α | k | τ , ∀K< | k | ≤ K 2.10 and Π Π\ R k .Then,∀p ∈ Π , it follows that kω p 2π − l ≥ α | k | τ , ∀ k, l ∈ × \ { 0, 0 } , 0 < | k | ≤ K , 2.11 where K > 0 such that e −K ρ E .Let T T 3 h ,h πα T K τ1 . 2.12 If h ≤ 2h/3,thenmax p∈Π h |ω p|≤T . Moreover, one has f s ,r ;h 1 r g s ,r ;h ≤ . 2.13 Thus, the above result also holds for A in place of A. Proof of Lemma 2.1. The above lemma is actually one KAM step. We divide the KAM step into several pats. (A) Truncation Let Q f fx, 0; p, Q g gx, 0; pg z x, 0; pz. It follows that Q f s,r;h ≤ , Q g s,r;h ≤ 2r. Write Q f k∈ Q fk pe ikx , Q g k∈ Q gk z; pe ikx ,andlet R f | k | ≤K Q fk p e ikx ,R g | k | ≤K Q gk z; p e ikx . 2.14 Advances in Difference Equations 7 By the definition of norm, we have Q f − R f s−ρ,r;h ≤ ce −Kρ , Q g − R g s−ρ,r;h ≤ ce −Kρ r. 2.15 (B) Construction of the Transformation As in 1–3, for a reversible mapping, if the change of variables commutes with the involution R, then the transformed mapping is also reversible with respect to the same involution R.If the change of variables U : ξ, η → x, z is of the form x ξ u ξ , z η v ξ, η , 2.16 then from the equality RU UR, it follows that u −ξ −u ξ , v −ξ, η v ξ, η . 2.17 In this case, the transformed mapping U −1 AU of A is also reversible with respect to the involution R : ξ, η → −ξ, η. In the following, we will determine the unknown functions u and v to satisfy the condition 2.17 in order to guarantee that the transformed mapping U −1 AU is also reversible. We may solve u and v from the following equations: u ξ ω p − u ξ R f ξ − R f ξ , v ξ ω p ,η − v ξ, η R g ξ, η − R g ξ, η , 2.18 where · denotes the mean value of a function over the angular variable ξ. Indeed, we can solve these functions from the above equations. But the problem is that such functions u and v do not, in general, satisfy the condition 2.17, that is, the transformed mapping U −1 AU is no longer a reversible mapping with respect to R. Therefore, we cannot use the above equations to determine the functions u and v. Instead of solving the above equations 2.18, we may find these functions u and v from the following modified equations: u ξ ω p − u ξ f ξ , v ξ ω p ,η − v ξ, η g ξ, η , 2.19 8AdvancesinDifference Equations with f ξ 1 2 R f ξ − R f ξ R f −ξ − ω p − R f −ξ − ω p , g ξ, η 1 2 R g ξ, η − R g −ξ − ω p ,η , 2.20 where · denotes the mean value of a function over the angular variable ξ. It is easy to verify that f−ξ − ωp fξ and g−ξ − ωp,η−gξ, η.So,by Lemma A.1, the functions u and v meet the condition 2.17. In this case, the transformed mapping U −1 AU is also reversible with respect to the involution R : ξ, η → −ξ, η. Because the right hand sides of 2.19 have the mean value zero, we can solve u, v from 2.19.Butthedifference equations introduce small divisors. By the definition of Π h ,it follows that ∀p ∈ Π h , kω p 2π − l ≥ α 2 | k | τ , ∀ k, l ∈ × \ { 0, 0 } , 0 < | k | ≤ K. 2.21 Let f k , g k be Fourier coefficients of f and g. Then, we have u k f k e ikωp − 1 ,v k g k e ikωp − 1 , 0 < | k | ≤ K, 2.22 and u k 0, v k 0fork 0or|k| >K.Moreover,v is affine in η, u is independent of η. (C) Estimates of the Transformation By the definition of norm, we have f s−ρ,r;h ≤ c, g s−ρ,r;h ≤ cr. 2.23 By Lemma A.1, it follows that u s−2ρ,r;h ≤ c αρ τ1 , v s−2ρ,r;h ≤ cr αρ τ1 . 2.24 Using Cauchy’s estimate on the derivatives of u, v,weobtain u ξ s−3ρ,r/2;h ≤ c αρ τ2 , v ξ s−3ρ,r/2;h < cr αρ τ2 , v η s−3ρ,r/2;h < c αρ τ1 . 2.25 Advances in Difference Equations 9 In the same way as in 1, 2, 4, we can verify that U −1 AU is well defined in Ds − 5ρ, μr, 0 <μ≤ 1/8. Moreover, according to 2.24–2.25,wehave W 0 U −id D s−5ρ,μr ×Π h ≤ c αρ τ2 , W 0 DU − Id W −1 0 D s−5ρ,μr ×Π h ≤ c αρ τ2 , 2.26 where ·denotes the maximum of the absolute value of the elements of a matrix, W 0 diagρ −1 0 ,r −1 0 , DU denotes the Jacobian matrix with respect to ξ, η. (D) Estimates of the New Perturbation Let α α − /2πK τ1 .Wehave|kω p/2π − l|≥α /|k| τ , ∀p ∈ Π, ∀0 < |k|≤K. Then, by the definition of R k , it follows that 2.11 holds. Thus, the small divisors condition for the next step holds. Let R f pR f ξ; p,thenwehaveR f p≤ αρ τ2 E.DuetoU −1 AU A ,we have f ξ, η u ξ − u ξ 1 − R f p f ξ u, η v . 2.27 By the first difference equation of 2.19,wehave f u ξ ω p − u ξ 1 f ξ u, η v − f ξ − R f p . 2.28 From the reversibility of A, it follows that f −x − ω p − f,z g − f x, z 0, g −x − ω p − f,z g g x, z 0. 2.29 Hence, we have f ξ, η − f ξ − R f p 1 2 f ξ, η − R f ξ f ξ, η − R f −ξ − ω p 1 2 f ξ, η − R f ξ f −ξ − ω p ,η − R f −ξ − ω p −f −ξ − ω p ,η f −ξ − ω p − f,η g , 2.30 which yields that f ξ, η − f ξ − R f p ≤ cμ ce −Kρ 2 2 ρ . 2.31 10 Advances in Difference Equations By 2.15 and 2.24–2.25, the following estimate of f holds: f ≤ u ξ · R f p f f ξ · u f η · v cμ ce −Kρ 2 2 ρ ≤ c αρ τ2 f c 2 αρ τ2 cμ ce −Kρ . 2.32 Similarly, for g ,weget g v ξ ω p ,η − v ξ 1 ,η 1 g ξ u, η v − g ξ, η , 2.33 1 r g ≤ c αμρ τ2 f c αρ τ1 g r c 2 αμρ τ2 cμ ce −Kρ μ . 2.34 If is sufficiently small such that c αμρ τ2 < 1 2 , 2.35 then combing with 2.32 and 2.34,wehave f s ,r ;h 1 r g s ,r ;h ≤ c 2 αμρ τ2 cμ ce −Kρ μ . 2.36 Suppose h ≤ 2/3h. Then, by Cauchy’s estimates, we have ω p − ω p ≤ 3 h , ∀p ∈ Π h . 2.37 Let T T 3/h. Then, max p∈Π h |ω p|≤T . Moreover, by the definition of ρ and E ,wehave f s ,r ;h 1 r g s ,r ;h ≤ c 2 αμρ τ2 ≤ cα ρ τ2 E 3/2 α ρ τ2 E . 2.38 Thus, this ends the proof of Lemma 2.1. Setting the Parameters and Iteration Now, we choose some suitable parameters so that the above iteration can go on infinitely. At the initial step, let ρ 0 1 − σs/10, r 0 r, 0 α 0 ρ τ2 0 E 0 .LetK 0 satisfy e −K 0 ρ 0 E 0 , α 0 α>0, ω 0 ω, T 0 T max p∈Π h |ω p|.Denote Π 0 p ∈ Π | kω p 2π − l ≥ α | k | τ , ∀0 < | k | ≤ K 0 . 2.39 [...]... 11001048 and the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers Grant no 200802861043 R Cheng was supported by the National Natural Science Foundation of China Grant no 11026212 References 1 B Liu, Invariant curves of quasi-periodic reversible mappings,” Nonlinearity, vol 18, no 2, pp 685– 701, 2005 2 B Liu and J J Song, Invariant curves of reversible mappings with... 20, no 1, pp 15–24, 2004 3 M B Sevryuk, Reversible Systems, vol 1211 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986 4 J Moser, “On invariant curves of area-preserving mappings of an annulus,” Nachrichten der Akademie der Wissenschaften in G¨ ttingen II Mathematisch-Physikalische Klasse, vol 1962, pp 1–20, 1962 o 5 C Simo, Invariant curves of analytic perturbed nontwist area preserving... extend V∗ as a Gevrey function of the same Gevrey index in a neighborhood of Π∗ Thus, by the definition of Gevrey function in a closed set, V∗ x, z; p ∈ G1,μ D s/2, r/2 × Π∗ , satisfies the estimate 1.13 and 1.15 in a neighborhood of Π∗ 16 Advances in Difference Equations Note that one can also use the inverse approximation lemma in 19 to prove the preceding Whitney extension for V∗ Estimates of Measure... extensions of differentiable functions defined in closed sets,” Transactions of the American Mathematical Society, vol 36, no 1, pp 63–89, 1934 17 J Bruna, “An extension theorem of Whitney type for non-quasi -analytic classes of functions,” The Journal of the London Mathematical Society, vol 22, no 3, pp 495–505, 1980 18 J Bonet, R W Braun, R Meise, and B A Taylor, “Whitney’s extension theorem for nonquasianalytic... 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We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be. and m is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves. 1. Introduction
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