EXISTENCE OF PERIODICALLY INVARIANT TORI ON RESONANT SURFACES FOR TWIST MAPPINGS

. In this paper we will prove the existence of periodically invariant tori of twist mappings on resonant surfaces under the low dimensional inter- section property.

which is composed of q 0 = lcm{q 1 , . . . , q m2 }, disjoint m 1 −dimensional tori that are linked by the iteration. In particular, if L is chosen such that ω 2 ∈ Z m2 , the mapping M possesses the invariant torus T m1 × {y 0 } × {z 0 } for any y 0 ∈ T m2 and z 0 ∈ O(D, L).
Consequently, instead of investigating the existence of periodically invariant tori of the mapping M, in this paper we are going to prove that the existence of invariant tori of the mapping M q0 . We make the following assumptions (H 1 ) : f, g, h, ω 1 are real analytic mappings when (x, y, z) ∈ T m1 ×T m2 ×O(D, L).
Throughout this paper, if we do not explicitly state, the norm of a matrix p = (p ij ) n1×n2 will be defined by where n 1 , n 2 ≥ 1 and p ij : D → C. Now we give some remarks. Firstly, since we are seeking for m 1 − dimensional invariant tori, the low dimensional intersection assumption (H 2 ) is reasonable. Secondly, when m 1 = m, that is, there is no any resonant relations in frequency vectors, Theorem 1.1 is the same as the corresponding theorem in [7], thus we generalize the result in [7] into the resonant case. Finally, from the proof of Theorem 1.1, if m 1 = n, that is, the number of the nonresonant frequencies ω 1 is equal to the number of the action variables, the nonresonant frequencies ω 1 can be kept unchanged under the iterative process, thus one can fix the nonresonant frequencies in advance and also do not need the measure estimate, just the same as the classical case in [21].
Before ending the introduction, let us recall some related results. In 1962, Moser [21] proved the existence of invariant closed curves of twist mappings with one angular variable and one action variable. When the number of angular variables is not equal to the number of action variables, the persistence of invariant tori can not be obtained by applying Moser's result in [21] directly. Cheng and Sun [5] in 1989 successfully proved that there exists a large set of two-dimensional invariant tori under certain nondegeneracy condition for three-dimensional measure preserving mappings. Xia [26] extended the result to the n + 1-dimensional volume-preserving diffeomorphisms with one action variable. Recently, Cong, Li and Huang [7] further extended the result to the mappings with distinct number of action variables and angular variables. They found that, under the Rüssmann condition and the intersection property, there also exists a large amount of invariant tori.
Since KAM theory was built by Kolmogorov [16], Arnold [1] and Moser [21] in 1960's, by now there have been many excellent works about the existence of resonant tori or lower dimensional invariant tori in Hamiltonian systems [4,10,11,8,22,25,27,17,18,19,3,23,20] and in symplectic mappings [2,28], as well as some results on the existence of invariant curves in quasi-periodic or almost periodic twist mappings [12,13,14,15], but the results about resonant tori of twist mappings are relatively few. Only Cheng and Sun [6] proved the existence of periodically invariant curves in 3-dimensional measure-preserving mappings. Simulated by the case of Hamiltonian systems, they made use of normal forms of 3-dimensional measurepreserving mappings and coped with the effect on the resonant frequency by seeking for the zeros of the average part of the perturbation about the resonant angular variable.
Different with the result in [6], on one hand, the mapping we are going to investigate has n action variables and m angular variables. On the other hand, we employ the completely different method to prove the existence of periodically invariant tori. Since the resonant angular variables are indeed slow variables on the resonant surface, one can use the low dimensional intersection property not only to deal with the average part of the perturbation about the action variables, which are also slow variables, but also to eliminate the average part of the perturbation about the resonant angular variables. More importantly, periodically invariant curves in [6] can be found only for the zeros of the average part of the perturbation about the resonant angular variables. While by the aid of the low dimensional intersection property, here we can obtain the existence of periodically invariant tori for any fixed resonant angular variables.
We organize the rest of the paper as follows. In Section 2, we will prove Theorem 1.1. The proof of the iterative Lemma is given in Section 3. In the last section, some lemmas used in the preceding sections are stated and proved.
2. The iterative lemma and the proof of theorem 1.1. The proof of Theorem 1.1 follows the traditional line laid out by [21] and [24]. The main step is to make a sequence of successive applications of the Iterative Lemma. In this section, we will state the lemma, and use it to prove Theorem 1.1.
The following four complex domains will make it convenient to state the lemma clearly, For a set X and a positive number r, define where ∂X is the boundary set of X.
Now we are in a position to state the iterative lemma.
with U 0 , V 0 , W 0 real analytic functions in A (1) of period 1 in ξ, η, such that the transformed mapping U −1 0 M q0 U 0 takes the form f (ξ, η, ζ)dξ, and f + , g + , h + are real analytic functions defined in B where they satisfy the estimate To be more exact: The proof of Lemma 2.1 will be provided in the next section. Now we will apply Lemma 2.1 to prove Theorem 1.1 on the existence of the periodically invariant tori. Before proving Theorem 1.1, it is necessary to construct the corresponding parameters of every iteration. For i = 0, 1, · · · , let where τ > L(L + 1) − 1 is assumed to be an integer and [·] denotes the integer part of a given real number.
Postponing the proof of the Lebesgue measure of the set G = ∞ j=0 O j to Subsection 2.2, we may assume G is a nonempty set and set According to Lemma 2.1, by the coordinate transformation U 0 , the perturbations of the resulting mapping M q0 1 can be made smaller than the original mapping M q0 , and the corresponding domain E 1 becomes narrower than E 0 . Making a sequence of successive applications of the lemma, we can have a sequence of coordinate transformations U i and the new mapping M q0 With the times of iterations increasing to infinity, the perturbations of the mapping M q0 i+1 approach to zero. Therefore, when the sequence { i } converges to 0, the mapping will be of the form Therefore, with respect to M q0 i , it is available to apply Lemma 2.1, thus we can obtain a coordinate transformation By Lemma 2.1 and Lemma 4.1, one obtain that which imply that and p i , q i , r i , ω 1 i are uniformly convergent in E ∞ . Denoting the limits of p i , q i , r i , and ω 1 i by u(ξ, η), v(ξ, η), w(ξ, η) − z 0 , and ω 1 ∞ (η, z 0 ), respectively, we find that a family of m 1 −dimensional periodically invariant tori are of the form Γ : , and the induced mapping on the tori Γ is expressed by This completes the proof of the existence of the periodically invariant torus in the mapping M. In the next subsection we will work on the measure estimate of the set G . This concludes the proof of Theorem 1.1. In the sequel, we will focus on the proof of Lemma 2.1.
3. The proof of the iterative lemma. In the process of the proof of Theorem 1.1 we made a sequence of successive applications of Lemma 2.1. Noticing that every iteration step is similar, without loss of generality, we only need to check one cycle of this iteration scheme. Here we just check the i th (i > 0, i ∈ N) step to the (i + 1) th step. From the following process we will find that the proof is also available for i = 0. Drop the indices "i" of corresponding parameters and variables, and replace the indices "i + 1" , "i − 1" with " + ", " − ", respectively. Write M q0 and      ξ q0 = ξ + ω 1 + (η, ζ) + f + (ξ, η, ζ), η q0 = η + g + (ξ, η, ζ), ζ q0 = ζ + h + (ξ, η, ζ).
Before proving the lemma, it will be convenient to introduce the following notations D (2) ={(x, y, z) : | x| < ρ + + 2δ, | y| < 8δ + + 2δ, Now we are in the position to prove Lemma 2.1. In fact, the first step of the proof of Lemma 2.1 is to find a coordinate transformation U. In order to obtain the coordinate transformation U, it is necessary to solve the following difference equations We solve the equations (9) by means of the Fourier expansion. Substituting into the third equation of (9), one obtains After obtaining the expression of W , it is necessary to prove that W is real analytic in (Σ 1 − δ) × (Σ 2 − δ) × G. Indeed, for 0 < |k| < N , there exists an integer k 0 satisfying that k 0 ≤ M 10 |k|, such that π| k, ω 1 (η, z 0 ) + k 0 | ≤ π/2.