Gevrey smooth topology is proper to detect normalization under Siegel type small divisor conditions

We shape the results on the formal Gevrey normalization. More precisely, we investigate the better expression of α^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\hat{\alpha }}}$$\end{document}, which makes the formal Gevrey-α^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\hat{\alpha }}}$$\end{document} coordinates substitution turning the Gevrey-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} smooth vector fields X into their normal forms in several cases. Such results show that the ‘loss’ of the Gevrey smoothness is not always necessary even under Siegel type small divisor conditions, which are different from others.


Introduction
The study of normal form theory has a long history, which is original from Poincaré. The basic idea is to simplify ordinary differential equations or diffeomorphisms through changes of variables near referenced solutions. Nowadays, the theory has extended its domain over various systems such as random dynamical systems, control systems and so on. Moreover, it also does great importance to bifurcations, stability theory and others.
As we have known, the celebrated Poincaré-Dulac scheme ensures the existence of formal normal forms. So the convergence of formal normal forms plays the central role of the whole research. Now let us recall some beautiful theorems on history. On the one hand, in Poincaré domain the system analytically conjugates to its polynomial normal forms. Meanwhile, in Siegel domain the system can be analytically linearized under some small divisor conditions. However, by the dichotomy method or the result of Yoccoz there actually exists a large gap between formal and analytic normal forms. On the other hand, in the rougher topology Hartman, Sternberg and Chen proved C 0 , C k and C ∞ conjugacy under the hyper-bolic condition, respectively. The brief introductions can be found in [1]. Anyway, above arguments remind us of the importance for the proper topology, where normal forms can inherit common properties from both analytic and C ∞ cases.
Then comes the Gevrey smooth topology in the ultra-differentiable class, that belongs to the C ∞ functional class but the derivatives of functions have certain norm controls. It can be regarded as the particular case in the ultra-differential class mentioned by Rudin [7]. With such a magnifying glass we persuade ourselves to detect their interactions of the classical Siegel small divisor conditions and non-vanished nonlinear resonant terms in the formal normal forms. More precisely, in the previous series work [2,5,6,9], related topics about Gevrey normalization were largely covered. Especially, restrict our focus on the classical vector fields X = Dx + R(x) in the Gevrey smooth category, where D is a diagonal matrix and R contains all higher order nonlinearities. It was proved in [9] (Theorem 1.11) when D = diag(λ 1 , . . . , λ d ) is hyperbolic and satisfies Siegel type conditions, i.e. it fulfills on all (k, j) ∈ nr = {(k, j) | |k| ≥ 2, k, λ − λ j = 0} for the positive constants c and μ, then the Gevrey-α smooth vector fields can be changed into their normal forms by the Gevrey-(α + μ + 1) smooth coordinates substitutions at the origin. Moreover, if X can be formally linearized additionally, then the conjugacy shall have no loss of Gevrey smoothness, namely, the changes persist Gevrey-α smooth. Such result is developed from [5], where it was proved that analytic vector can be changed into their normal forms via the formal Gevrey 1 + τ transformations, and then [6] for more degenerated vector fields and formal Gevrey α vector fields. See also Sternberg's pioneering work [2] and [8] for the hyperbolic smooth and Gevrey linearization, respectively. Therefore, the natural gap between α + μ + 1 and α implies the possibility of the existence of the fine structures for nonlinear resonant terms.
In this way, we consider the Gevrey-α smooth system where D is diagonal, N is nilpotent and R contains all higher order nonlinearities. By taking its formal normal forms, it admits ] is a formal Taylor series, then q is denoted by the lowest degree of terms in N x +R. Of course, the formal linearization corresponds to the procedure as q → ∞. And q = 1 is in general the worst case if D = 0, which prevents the convergence of formal transformations frequently. Next we have the following two conditions for system (1) The condition (C1) stems from the Poincaré domain. Whereas condition (C2) implies the restriction q ≥ 2. When μ > 0, it accords with the classical Siegel small divisor condition. When μ = 0, it is satisfied by complete integrable systems from [10]. If −1 ≤ μ < 0, we have polynomial formal normal forms in general. In this paper, our results can be summarized as follows. At this moment, the study of Gevrey smooth normal forms all follows Stolovitch's two steps scheme. It begins with the seeking of formal Gevrey smooth normal forms, which provides a necessary aim for the further exploration. Then for the 'real' Gevrey smooth system we only need deal with the cancelation of Gevrey flat remainders due to Gevrey Whitney type extension theorems. When the system is hyperbolic specially, the Gevrey smoothness of the transformation can be directly checked in a complicated but explicit formula as shown in [9]. In this paper, we mainly improve the results to get an accurate expression of the Gevrey indexα at the first step. Comparing with other results, On the one hand, from μ+1 q−1 → 0 as q → ∞, it precisely characterizes the action of resonant terms on the convergence of changes. On the other hand, it implies that the increasing of the Gevrey index is not always necessary in the normalization even under Siegel small divisor conditions, which strengths the result in [9]. Above all we think that those clearly indicate the effect of different topology on the normalization. Now we consider system (1) under Siegel type small divisor conditions. For the analytic topology, there is no analytical normalization as α = 0. Now the topology changes weak as Gevrey smooth index α increases. When the index is small, it happens the loss of Gevrey smoothness, i.e. the convergent transformation has larger indexα = μ+1 q−1 than α < μ+1 q−1 . However, when the Gevrey index is large enough, the loss stops forα = α ≥ μ+1 q−1 . Until the weakest C ∞ topology, the normalization is always guaranteed in the hyperbolic case. But when q → ∞ as the boundary value case, such slight difference disappears and formal convergence is always valid. So we say that the topology of Gevrey smoothness is proper to detect fine structures of the normalization under Siegel type conditions.
The rest paper are organized as follows. In Sect. 2, notations, definitions and basic lemmas are written down. Then in Sect. 3, we solve the homological equation, which is the linear approximation of the equation given by normal form reductions. So in Sect. 4, KAM methods and Contracting Mapping Principle can be applied to get formal Gevrey normalization for μ ≥ 0 and μ < 0, respectively. Thus, together with Stolovitch's arguments we get our main theorem in the last section.

Set
3. Use f r = (k, j)∈ r f k, j x k e j for the given f = |k|≥1, j f k, j x k e j . 4. Use f nr = (k, j)∈ nr f k, j x k e j for the given f = |k|≥2, j f k, j x k e j . 5. As usual, ∂ k f is the k-th order differential operator.
Then we introduce some basic information about Gevery type smoothness. Let be an open set R d and α ≥ 1. A smooth complex-valued function f on this set is said to be Gevery-α smooth, if for any compact set K ⊆ , there exist positive constants M and C such that As usual, ∂ k f is the k-th order differential operator. However, the formal power series f = k, j f k, j x k e j ∈ C d [[x]] is said to be formal Geverey-α if there exist positive constants M and C such that | f k, j | ≤ MC |k| (|k|!) α . Of course, we shall note that the Taylor expansion at the origin of a smooth Gevrey-α function is a formal Gevrey-(α + 1) power series. See [9] for more details. Hence we modify the majorant norm. For any formal power series f ∈ C d [[x]] and the fixed α ≥ 0, we can set Proof First we show a fact. Notice that |l|! ≥ t! t i=1 (l i !) by setting l = (l 1 , . . . , l t ) ∈ Z t + satisfying l i > 0 for i = 1, . . . , t. If l i = 1 for i = 1, . . . , t or t = 1, by simple computations we have the equality. Otherwise, it yields |l| > max{t, l i }. Since in |l|! there are |l| − 1 factors except the trivial number 1 and the same umbers which leads to Since we have that |m (i, j) | ≥ 1 for all i and j by f (0) = 0, by the above fact it leads to by Lemma 2(i). This completes the proof.
With the aid of the above corollary, the preparation for the study of modified majorant norms is ready.

Proposition 4
The following statements hold for the modified majorant norm | · | r,α .
The general case of an arbitrary r follows from the fact the correspondence Then we verify results (ii) and (iii). On the one hand, by simple computations we obtain that That completes the proof.
Next for any f ∈ C d [[x]] we define the power shift operator P μ (μ ≥ −1) given by Then we study the property of P μ acting on the classical differential type operator ∂ x i with respect to the variable x = (x 1 , . . . , x d ), which is the key of the whole proof.

Lemma 5
The following statements hold.
Here the positive constant C(α, q) is given by depending on α and q only. Proof First we note that u! ≥ s!t! for u + 1 = s + t, s ≥ 1 and t ≥ 1. Since t ≥ 1 and s ≥ 1 imply u ≥ s and u ≥ t, by counting the non-trivial factor except number 1 of both sides, it yields u − 1 = s − 1 + t − 1, which confirms our result. Then by simple calculations, we obtain that This verifies (i).
To confirm (ii), we verify that for s + t = u + 1, s ≥ q and t ≥ q at first. If s ≥ t, then we have that For the case s < t, we get that s q−1 s!t! ≤ (t + q − 1)!(u + 1 − t)! and other arguments are similar. Then for μ ≥ 0, from (4) we can show that where Therefore, by setting |k| ≥ q, |l| ≥ q and α ≥ μ+1 q−1 , when μ ≤ 0, from (4) we obtain that And when μ > 0, from (5) we have that This completes the proof.

The solution of the homological equation
In this part, we discuss the formal and Gevrey smooth solvable of the homological equation where On the one hand, from the fact On the other hand, since the linear nilpotent part N is in the lower triangle form, then we can define a full order < on nr , which in given by (k, j) < (k , j ) for |k| < |k |, or |k| = |k |, but j < j or k s < k s with j = j and s = min{t | k t = k t }. Arbitrary choosing one monomial x l e m for (l, m) ∈ nr , then it leads to and others monomials are in P 2 . Therefore, the part P 1 + P 2 contains all monomials whose indexes are larger than (l, m). Namely, we can solve (6) by this full order on the index set nr . This completes the proof.
Set D = diagλ. Here we restrict the focus on two cases of equation (6) under conditions (C1) and (C2). Denote by ad F H = [F, H ]. If F = Ax is linear, we simply use ad A instead of ad Ax . Let F (s) = |k|=s, j F k, j x k e j for F = k, j F k, j x k e j . And set ord(F) = min{|k| | F k, j = 0, |k| > 1, (k, j) ∈ r }. Moreover, in the next lemma we can get δ = 0 for μ ≤ 0 and δ > 0 for μ > 0 in result (ii). Thus by using the remark 0 μ = 1, we can get the uniform expression.

Proposition 7
The following statements hold for the solution of equation (6).
Here ad −1 D (G) denotes the unique solution H satisfying H = H nr to ad D (H ) = G, which has a clear representation Under condition (C1), without loss of generality we can assume that the linear nilpotent part has the form εN , where entries of N are 1 or 0. Thus On the one hand, by Lemma 5(ii) it implies where P μ is the same operator given by (2). On the other hand, by the fact we obtain that |N H| r,α ≤ ε|H | r,α and When ε ≤ c/4 and r 1 ≤ r 2 0 /(4C 1 |F| α,r 0 ), we obtain that |ad −1 D • ad R | ≤ 3/4, then (Id + ad −1 D • ad R ) has an inverse given by the Neuman series with the control |(Id Then comes the case (C2). Set F (s) and H (t) be the homogeneous polynomials of degree s and t, respectively. From Proposition 6, the solution of (6) formally exists, which yields H = t≥q H (t) by comparing terms of lowest degree on both sides. Using more precise estimations, first by Lemma 2 we obtain that ξ . However, note that we have that Thus we can solve (6) by using the expansion F = Dx + s≥q F (s) and H = s≥q H (s) for G = s≥q G (s) . Naturally, the solution is governed by By the estimation (7) we obtain that Then with the restriction α ≥ μ+1 q−1 , for μ > 0 it admits from (5) and for μ ≤ 0 it leads to from (4). Now set σ = re −δ and σ = r for μ > 0 and −1 < μ ≤ 0, respectively. When μ > 0, we obtain that While μ ≤ 0, then u μ ≤ 1 and it yields Here C 2 = max{c −1 μ μ , c −1 } and C 3 = 2c −1 C(α, q) from the fact max x≥0 {x μ e −δx } ≤ μ μ δ −μ . Now choosing a very large N , summing all inequalities together we obtain that Note that we have δ > 0 for μ > 0 and δ = 0 for −1 < μ ≤ 0. If we make 0 μ = 1, then it leads to the same expression. Making N → ∞, we get that |H | α,re −δ ≤ 2C 2 δ −μ |G| r,α , when 2C 3 r −1 e δ |F| re −δ ,α < 1. This completes the proof.

KAM methods and contracting mapping principle in the formal Gevrey normalization
In this part, we use KAM steps and Contracting Mapping Principle to detect formal Gevrey normalization for μ < 0 and μ ≥ 0, respectively. Here we follow the scheme as shown in [3] (pp. 70-72) and [4] (pp. 52-56) by some modifications due to our case. First we take the case μ ≥ 0 into account.

Consider the systemẋ
where D = diagλ, f nr = f nr . Without loss of generality, we can set that the degree of monomials in f nr is greater than q. Otherwise, we can apply the Poincaré-Dulac formal normal form reductions to do cancelations. Doing the coordinates substitution x = y + h(y) with h nr = h to system (8), it yieldṡ where and First we study the remainder parts R 1 and R 2 .

Proposition 8 Assume that f and h ∈ C d [[x]]
and h(0) = 0. Set ρ = r + |h| r,α . Then we have that Proof This proof shares the same kernel as Proposition 4. For the fixed t, k = (k 1 , . . . , k d ) and h = (h 1 , . . . , h d ), using the equality and by rough estimations we obtain that from the classical mean value inequality, where C t q = q! (q−t)!t! . Then from Corollary 3 we obtain that by the fact max x≥0 {xe −δx } ≤ δ −1 . Furthermore, by similar arguments we obtain that where C = 4 by the same arguments. That completes the proof. By Proposition 7, we takeĥ to be the non-resonant solution of [Dy + f r ,ĥ] = − f nr in system (9), which leads to a new onė Note that ord(ĥ) ≥ q. Now comes the iterative lemma.
Thus the formal coordinates substitution can be found for μ ≥ 0.
Then we deal with the case −1 ≤ μ < 0 by Contracting Mapping Principle. Since the formal normal form is a polynomial by Proposition 15, we consider the particular form of system (1) as followsẋ where P and R are nonlinearities satisfying P is a polynomial, P nr = 0 and R nr = R, N is the well chosen nilpotent linear part fulfilling Proposition 7(i) for μ = −1 and N = 0 for −1 < μ < 0. Without loss of generality, we can assume that the degree of all nonlinear monomials in R is greater thanq = deg(P). As usual, deg(P) is the degree of the polynomial P. If the transformation x = y + h(y) can turn system (20) into its normal forṁ y = (D + N )y + P(y), where F(y) = (D + N )y + P(y) and [·, ·] is the classical Lie bracket with respect to the variable y. Now we restrict our focus on the ball equipped with the norm | · | r,α , where s =q + 1 ≥ 2 for μ = −1 and s =q + 1 ≥ q ≥ 2 for −1 < μ < 0. Hereq and q are the same defined as before. Then for any operator T acting on the formal vector series h, we say that the operator T is strongly contracting, if |T (0)| r,α = O(r 2 ) and T is Lipschitz on the ball B r under the norm | · | r,α , with the Lipschitz constant no greater than O(r ) as r → 0. As usual, O(1) refers to the bounded quantity by a limiting process. In this context, denote operators T 1 , T 2 and T 3 by Hence, equation (21) has an equivalent form by above operators Next come the properties of T i for i = 1, 2 and 3.
Proof First we note again that provided that r < r 0 , g = |k|≥s, j g k, j x k e j ∈ C d [[x]] and |g| r 0 ,α < ∞. Since we have set that the degree of all nonlinear monomials in R is greater thanq = deg(P), so is the degree of ones in T i for all i. Make s =q + 1 ≥ 2 for μ = −1 and s =q + 1 ≥ q ≥ 2 for −1 < μ < 0 as above. Then, by Lemma 5(i) and the above fact, the linear operator T 1 satisfies where C 4 = (ln 2) −1 r −s 0 from Lemma 5(i) and r ≤ r 0 /2. Whatever the case is, it leads to s − 1 ≥ 1 and we have the strongly contractive operator T 1 .
With the aid of above lemma and Proposition 7 we can solve (22) finally.
Proof As we have shown, the existence of the change is equivalent to the solvability of the operator equation (22). Rewrite it in another form, (22) turns to , where ad F (·) = [F, ·] and T i is the same as defined above for i = 1, 2, 3. Notice that no resonance happens in B r . Therefore, by Proposition 7(i) the operator ad −1 F is bounded for μ = −1. Note that |P| r,α = O(r q ) as r → 0. So the condition of Proposition 7(ii) is also satisfied, which means ad −1 F is bounded for −1 < μ < 0, provided that we take r small enough. Then from Lemma 11 the operators T i is strongly contractive for i = 1, 2, 3. And so is ad −1 F • (T 1 − T 2 − T 3 ). Thus, we can chooser > 0 small enough such that ad −1 F • (T 1 − T 2 − T 3 ) maps Br into itself and the corresponding Lipschitz of this operator is less then 1. By Contracting Mapping Principle, we completes the proof.

Proof of the main theorem
In this part, we provide the proof of the main theorem and do further considerations.
At last, we consider one known result in our context, which refers to Bruno type conditions(Proposition 2.5, pp. 248) in [9] under the assumption that the system can be formally linearized. Now altering the classical Bruno conditions into the small divisor form, our methods can be applied.
which has the similar form as equation (24) for d = d 0 − 1 and using A − tλ d 0 instead of A. Therefore, it completes the proof by the induction method of the second type. This completes the proof.
Then comes the research of the set r . As usual, denotes the number of the points in the set . Here the sets r and nr are same as before.
Proof Assume that r = ∞. Without loss of generality, we can set that equation has infinitely many solutions for |k| ≥ 2 and k ∈ Z d + . Thus by Lemma 14 there existsk ∈ Z d + such that k , λ = 0 and |k| > 0. On the one hand, arbitrary choosing ( k, j) ∈ nr for nr = ∅, we have that = (tk + k, j) | t ∈ Z + ⊆ nr from the fact tk + k, λ − λ j = t k , λ + k, λ − λ j = k, λ − λ j = 0 for any t. On the other hand, form the large divisor condition it yields Making t → ∞, the right side of the above inequalities shall turn to infinite by μ < 0. But the left side is a constant for all t, which leads to a contradiction. This completes the proof.