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Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings
Advances in Difference Equations volumeÂ 2010, ArticleÂ number:Â 324378 (2010)
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 :
where the rotation is real analytic and satisfies the weaker nondegeneracy condition
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 nondegeneracy conditions are satisfied and , are sufficiently small, the existence of invariant curve of reversible mapping (1.1) has been proved in [1â€“3]. For related works, we refer the readers to [4â€“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 PÃ¶schel [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 RÃ¼ssmann's nondegeneracy condition [10â€“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 nondegeneracy 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
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 Gevreyclass of index if is smooth and there exists a constant such that for all ,
where and for .
Remark 1.2.
By definition, it is easy to see that the Gevreysmooth functions class coincides with the class of analytic functions. Moreover, we have
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
and denote a complex neighborhood of by
Now the function is real analytic on . We expand as Fourier series with respect to
then define
where
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 nondegeneracy condition: , , . Suppose that and are real analytic on . Then, there exists such that for any , if
there is a nonempty Cantor set , and a family of transformations , ,
satisfying , and for any ,
where , the constant depends on , , and . Under these transformations, the mapping (1.3) is transformed to
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
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 nondegeneracy 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 PÃ¶schel 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 :
on . Suppose the mapping is reversible with respect to the involution , that is, . Let , , and such that . Suppose , the following small divisors condition holds:
Let
Suppose that
where the norm indicates for simplicity. Then, for any , there exists a transformation :
which is affine in , such that the mapping is transformed to :
where the new perturbation satisfies
with
where is defined in Theorem 1.4. Moreover, one has
Let , and denote
and . Then, , it follows that
where such that . Let
If , then . Moreover, one has
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
By the definition of norm, we have

(B)
Construction of the Transformation
As in [1â€“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
then from the equality , it follows that
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:
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:
with
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 ,
Let , be Fourier coefficients of and . Then, we have
and , for or . Moreover, is affine in , is independent of .

(C)
Estimates of the Transformation
By the definition of norm, we have
By Lemma A.1, it follows that
Using Cauchy's estimate on the derivatives of , , we obtain
In the same way as in [1, 2, 4], we can verify that is well defined in , . Moreover, according to (2.24)â€“(2.25), we have
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
By the first difference equation of (2.19), we have
From the reversibility of , it follows that
Hence, we have
which yields that
By (2.15) and (2.24)â€“(2.25), the following estimate of holds:
Similarly, for , we get
If is sufficiently small such that
then combing with (2.32) and (2.34), we have
Suppose . Then, by Cauchy's estimates, we have
Let . Then, .
Moreover, by the definition of and , we have
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
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
Denote and for simplicity. By the iteration lemma, we have a sequence of transformations :
such that for any , , satisfying
where , â€‰ denotes the Jacobian matrix with respect to .
Thus, the transformation is well defined in and is seen to take into
More precisely, if we write as
and express in the form
then is transformed into :
satisfying
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
If is sufficiently small such that
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
In the same way as in [4, 7], we have
where denotes the maximum of the absolute value of the elements of a matrix.
By Cauchy's estimate we have
Let and . By and the definition of , we have
where . It is easy to see that
Thus, we have
where , depends on , , and .
In the same way, we have
Note that , , , as . Let , and . Since is affine in , these estimates (2.52)â€“(2.56) imply that is uniformly convergent to on and satisfies
Since , this proves (1.13).
Let . It follows that
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
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
For , we have
Then we obtain as above
and we get
where , â€‰ depends on , , and . Similarly, for , we obtain the same inequality.
Let and . According to (2.64), the limit satisfies that
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
where
with .
By Lemma A.2, we have
Since , we have
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Acknowledgments
We would like to thank the referees very much for their valuable comments and suggestions. D. Zhang was supported by the National Natural Science Foundation of China Grants nos. (10826035)(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).
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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:
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
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
where .
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Zhang, D., Cheng, R. Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings. Adv Differ Equ 2010, 324378 (2010). https://doi.org/10.1155/2010/324378
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DOI: https://doi.org/10.1155/2010/324378