- Research Article
- Open Access
Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings
© D. Zhang and R. Cheng. 2010
- Received: 19 April 2010
- Accepted: 25 December 2010
- Published: 29 December 2010
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.
- Invariant Curve
- Reversible Mapping
- Invariant Curf
- Small Divisor
- Gevrey Class
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 , 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  obtained Gevrey smoothness of invariant tori for analytic Hamiltonian systems. In , 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.
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.
where and for .
In this paper, we will prove Gevrey smoothness of function in a closed set, so we give the following definition.
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 .
We write if is analytic with respect to on and -smooth in on .
Denote . Fix and , and let and . Let .
Moreover, one has .
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.
The derivatives in (1.13) and (1.15) should be understood in the sense of Whitney . In fact, the estimates (1.13) and (1.15) also hold in a neighborhood of with the same Gevrey index.
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  by Pöschel and adapted to Gevrey classes in  by Popov. We also extend the method of Liu [1, 2].
The KAM step can be summarized in the following lemma.
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.
Construction of the Transformation
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.
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 .
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 .
and , for or . Moreover, is affine in , is independent of .
Estimates of the Transformation
where denotes the maximum of the absolute value of the elements of a matrix, , denotes the Jacobian matrix with respect to .
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, .
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.
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.
where , denotes the Jacobian matrix with respect to .
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 .
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
where denotes the maximum of the absolute value of the elements of a matrix.
where , depends on , , and .
Since , this proves (1.13).
Moreover, we have , , , where with . Thus (1.15) and (1.16) hold.
Whitney Extension in Gevrey Classes
where , depends on , , and . Similarly, for , we obtain the same inequality.
Since satisfies (2.58) and (2.65), by Theorem 3.7 and Theorem 3.8 in , 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  to prove the preceding Whitney extension for .
Estimates of Measure for Parameters
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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