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This paper develops a new KAM theorem for a class of lower dimensional elliptic invariant tori of nearly integrable symplectic mappings with generating functions but without assuming any nondegenerate condition.

The research on nondegeneracy condition of Hamiltonian systems is the fundamental problem of KAM theory because of small divisor problem. There are a significant number of results on nondegeneracy condition. We refer to Kolmogorov [

As one part of the classical KAM theory, the persistence of invariant tori of nearly integrable twist mappings was investigated by lots of mathematicians. The first work was due to Moser [

Motivated by [

We consider the following parameterized symplectic mapping

There are a great number of results in symplectic mappings and Hamiltonian systems, which are parallel and almost identical, but the proofs are different. The reason lies in special properties of symplectic mappings. For instance, the generating functions, which decide the symplectic mappings and the relation of variables, take on sophisticated implicit forms. So the relation of variables in symplectic mappings is not easily understood, which makes KAM estimates more complicated.

Without assuming any nondegenerate condition, we will give a formal KAM theorem for symplectic mappings. The idea of the proof is to separate the nondegenerate conditions from the KAM iteration, which was introduced in [

Before giving the main result, we introduce some assumptions.

Set

Consider the symplectic mapping

Suppose

Set

Consider the parameterized symplectic mapping

(i) There exist a family of symplectic mappings

(ii) If

By Theorem

Consider the symplectic mapping

(i)

(ii) Let

For the convenience, let

We need a symplectic transformation

By (

The conjugate symplectic mapping

Moreover, set

Now we will solve homological equations for

We will use the idea in [

Let

Firstly, for

Next we try to get

To get the relations between

Thirdly, we solve the third equation of (

Let

To get the relations between

Then we have

This part is critical for this paper. Let

Firstly, we extend

Secondly, we extend

At last, we expend

By the above discussion, we get a function

From the above discussion, we get a conjugate mapping

We note that the normal form depends only on

There exists a symplectic mapping

We note that

Let

We aim at the estimate of

By (

We choose a weighted error

By (

In this section, we will summarize the above results on parameters so that KAM-step can iterate infinitely. At the initial step set

Similar to the proof of KAM iteration Lemma, we have

Now we prove the convergence of KAM iteration. In the same way as [

Let

Correspondingly, let

Let

The author declares that he has no competing interests.

The paper was completed during the author’s visit to Department of Mathematics of Pennsylvania State University, supported by Nanjing Tech University. The author thanks Professor Mark Levi for his inviting, hospitality, and valuable discussions. The work is supported by Natural Science Foundation of Jiangsu Higher Education Institutions of China (14KJB110009).