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.
1. Introduction and Main Results
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.
In this paper, we will prove Gevrey smoothness of function in a closed set, so we give the following definition.
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.
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  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.
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
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 .
Estimates of the Transformation
Estimates of the New Perturbation
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.
Whitney Extension in Gevrey Classes
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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