Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings

Abstract

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 , where is the exponent in the small divisors condition and 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 :

(1.1)

where the rotation is real analytic and satisfies the weaker non-degeneracy condition

(1.2)

where and are real analytic and periodic in , the variable ranges in an open interval of the real line . We suppose that the mapping is reversible with respect to the involution , that is, . When some nonresonance and non-degeneracy conditions are satisfied and , are sufficiently small, the existence of invariant 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 mathematicians turn to study the regular property of KAM tori with respect to parameters. One of the earliest results is due to Pöschel [7], who proved that the KAM tori of nearly integrable analytic Hamiltonian systems form a Cantor family depending on parameters only in -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 -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 Rüssmann'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 , the existence of a -family of invariant curves has been proved in [1, 2]. But in the case of weaker non-degeneracy condition (1.2), there is no result about Gevrey smoothness. In this paper, we are concerned with Gevrey smoothness of invariant curve of reversible mapping (1.1). The 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 , and expand around , so that , where , , varies in a neighborhood of origin of the real line . We put , , and obtain the family of reversible mappings

(1.3)

Now, we turn to consider this family of reversible mappings with parameters , where 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 be a domain of . A function is said to belong to the Gevrey-class of index if is -smooth and there exists a constant such that for all ,

(1.4)

where and for .

Remark 1.2.

By definition, it is easy to see that the Gevrey-smooth functions class coincides with the class of analytic functions. Moreover, we have

(1.5)

for .

In this paper, we will prove Gevrey smoothness of function in a closed set, so we give the following definition.

Definition 1.3.

A function is Gevrey of index on a compact set if it can be extended as a Gevrey function of the same index in a neighborhood of .

Define

(1.6)

and denote a complex neighborhood of by

(1.7)

Now the function is real analytic on . We expand as Fourier series with respect to

(1.8)

then define

(1.9)

where

(1.10)

We write if is analytic with respect to on and -smooth in on .

Denote . Fix and , and let and . Let .

Theorem 1.4.

We consider the mapping defined in (1.3), which is reversible with respect to the involution , that is, . Suppose that satisfies the non-degeneracy condition: , , . Suppose that and are real analytic on . Then, there exists such that for any , if

(1.11)

there is a nonempty Cantor set , and a family of transformations , ,

(1.12)

satisfying , and for any ,

(1.13)

where , the constant depends on , , and . Under these transformations, the mapping (1.3) is transformed to

(1.14)

where , at . Thus, for any , the mapping (1.3) has an invariant curve such that the induced mapping on this curve is the translation , whose frequency satisfies that

(1.15)

(1.16)

Moreover, one has .

Remark 1.5.

From Theorem 1.4, we can see that for any , if is sufficiently small, the family of invariant curves is -smooth in the parameters. The Gevrey index 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 with the same Gevrey index.

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 Pöschel 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 :

(2.1)

on . Suppose the mapping is reversible with respect to the involution , that is, . Let , , and such that . Suppose , the following small divisors condition holds:

(2.2)

Let

(2.3)

Suppose that

(2.4)

where the norm indicates for simplicity. Then, for any , there exists a transformation :

(2.5)

which is affine in , such that the mapping is transformed to :

(2.6)

where the new perturbation satisfies

(2.7)

with

(2.8)

where is defined in Theorem 1.4. Moreover, one has

(2.9)

Let , and denote

(2.10)

and . Then, , it follows that

(2.11)

where such that . Let

(2.12)

If , then . Moreover, one has

(2.13)

Thus, the above result also holds for in place of .

Proof of Lemma 2.1.

The above lemma is actually one KAM step. We divide the KAM step into several pats.

• (A) Truncation

Let , . It follows that , . Write , , and let

(2.14)

By the definition of norm, we have

(2.15)

• (B) Construction of the Transformation

As in [1–3], for a reversible mapping, if the change of variables commutes with the involution , then the transformed mapping is also reversible with respect to the same involution . If the change of variables is of the form

(2.16)

then from the equality , it follows that

(2.17)

In this case, the transformed mapping of is also reversible with respect to the involution .

In the following, we will determine the unknown functions and to satisfy the condition (2.17) in order to guarantee that the transformed mapping is also reversible.

We may solve and from the following equations:

(2.18)

where denotes the mean value of a function over the angular variable . Indeed, we can solve these functions from the above equations. But the problem is that such functions and do not, in general, satisfy the condition (2.17), that is, the transformed mapping is no longer a reversible mapping with respect to . Therefore, we cannot use the above equations to determine the functions and .

Instead of solving the above equations (2.18), we may find these functions and from the following modified equations:

(2.19)

with

(2.20)

where denotes the mean value of a function over the angular variable .

It is easy to verify that and . So, by Lemma A.1, the functions and meet the condition (2.17). In this case, the transformed mapping is also reversible with respect to the involution .

Because the right hand sides of (2.19) have the mean value zero, we can solve , from (2.19). But the difference equations introduce small divisors. By the definition of , it follows that ,

(2.21)

Let , be Fourier coefficients of and . Then, we have

(2.22)

and , for or . Moreover, is affine in , is independent of .

• (C) Estimates of the Transformation

By the definition of norm, we have

(2.23)

By Lemma A.1, it follows that

(2.24)

Using Cauchy's estimate on the derivatives of , , we obtain

(2.25)

In the same way as in [1, 2, 4], we can verify that is well defined in , . Moreover, according to (2.24)–(2.25), we have

(2.26)

where denotes the maximum of the absolute value of the elements of a matrix, , denotes the Jacobian matrix with respect to .

• (D) Estimates of the New Perturbation

Let . We have , , . Then, by the definition of , it follows that (2.11) holds. Thus, the small divisors condition for the next step holds.

Let , then we have . Due to , we have

(2.27)

By the first difference equation of (2.19), we have

(2.28)

From the reversibility of , it follows that

(2.29)

Hence, we have

(2.30)

which yields that

(2.31)

By (2.15) and (2.24)–(2.25), the following estimate of holds:

(2.32)

Similarly, for , we get

(2.33)

(2.34)

If is sufficiently small such that

(2.35)

then combing with (2.32) and (2.34), we have

(2.36)

Suppose . Then, by Cauchy's estimates, we have

(2.37)

Let . Then, .

Moreover, by the definition of and , we have

(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 , , . Let satisfy , , , . Denote

(2.39)

Choose . Note that this choice for is only for measure estimate for parameters and has no conflict with the assumption in Theorem 1.4, since we can use a smaller .

Let and . Assume the above parameters are all well defined for . Then, we define , and , . Define , , , and in the same way as the previous step.

Let

(2.40)

Denote and for simplicity. By the iteration lemma, we have a sequence of transformations :

(2.41)

such that for any , , satisfying

(2.42)

where ,   denotes the Jacobian matrix with respect to .

Thus, the transformation is well defined in and is seen to take into

(2.43)

More precisely, if we write as

(2.44)

and express in the form

(2.45)

then is transformed into :

(2.46)

satisfying

(2.47)

In the following, we will check the assumptions in the iteration lemma to ensure that KAM step is valid for all .

Since and , if is sufficiently small such that , it follows that . Thus .

By the definition of , we have

(2.48)

If is sufficiently small such that

(2.49)

then we obtain and so , .

Obviously, if is sufficiently small, the assumption (2.35) holds.

By , it is easy to see that . If is sufficiently small and so is sufficiently large such that , then we have , .

Suppose . Let . Then, we have .

By iteration, . Suppose , then we have . If is sufficiently small such that , then .

Convergence of Iteration in Gevrey Space

Now, we prove convergence of KAM iteration. Let , and write in the form

(2.50)

In the same way as in [4, 7], we have

(2.51)

where denotes the maximum of the absolute value of the elements of a matrix.

By Cauchy's estimate we have

(2.52)

(2.53)

Let and . By and the definition of , we have

(2.54)

where . It is easy to see that

(2.55)

Thus, we have

(2.56)

where , depends on , , and .

In the same way, we have

(2.57)

Note that , , , as . Let , and . Since is affine in , these estimates (2.52)–(2.56) imply that is uniformly convergent to on and satisfies

(2.58)

Since , this proves (1.13).

Let . It follows that

(2.59)

Moreover, we have , , , where with . Thus (1.15) and (1.16) hold.

Whitney Extension in Gevrey Classes

In this section, we apply the Whitney extension theorem in Gevrey classes [13, 17, 18] to extend as a Gevrey function of the same Gevrey index in a neighborhood of .

Denote , then for any positive integers , , and with , we denote

(2.60)

In order to apply the Whitney extension theorem in Gevrey classes for function , we are going to estimate . First, we suppose that , . Expanding in the analytic function , , in Taylor series with respect to at , and using the Cauchy estimate, we evaluate

(2.61)

For , we have

(2.62)

Then we obtain as above

(2.63)

and we get

(2.64)

where ,   depends on , , and . Similarly, for , we obtain the same inequality.

Let and . According to (2.64), the limit satisfies that

(2.65)

Since satisfies (2.58) and (2.65), by Theorem 3.7 and Theorem 3.8 in [13], we can extend as a Gevrey function of the same Gevrey index in a neighborhood of . Thus, by the definition of Gevrey function in a closed set, , satisfies the estimate (1.13) and (1.15) in a neighborhood of .

Note that one can also use the inverse approximation lemma in [19] to prove the preceding Whitney extension for .

Estimates of Measure for Parameters

Now we estimate the Lebesgue measure of the set , on which the small divisors condition holds in the KAM iteration. By the analyticity of and , , for almost all points in , . Without loss of generality, we suppose , . Then, by the KAM step, we have

(2.66)

where

(2.67)

with .

By Lemma A.2, we have

(2.68)

Since , we have

(2.69)

Appendix

A. Some Results on Difference Equation and Measure Estimate?

In this section, we formulate some lemmas which have been used in the previous section. For detailed proofs, we refer to [1, 20].

In the construction of the transformation in Lemma 2.1, we will meet the following difference equation:

(A1)

Lemma A.1.

Suppose that l(x), g(x) are real analytic on , where . Suppose satisfies the Diophantine condition , , then for any , the difference equation (A.1) has the unique solution satisfying

(A2)

Moreover, if , then is odd in if , is even in .

Lemma A.2.

Suppose is mth differentiable function on the closure of , where is an interval. Let , . If for all , where is a constant, then

(A3)

where .