proved a KAM theorem on lower dimensional elliptic invariant tori for nearly integrable symplectic mappings.
Motivated by [9, 18, 22], we will extend the result  to symplectic mappings and prove a KAM theorem of symplectic mappings with generating functions but without assuming any nondegenerate condition.
Without assuming any nondegenerate condition, we will give a formal KAM theorem for symplectic mappings.
Xu, "A KAM theorem for a class of nearly integrable symplectic mappings," Journal of Dynamics and Differential Equations, pp.
Shang, "A note on the KAM theorem for symplectic mappings," Journal of Dynamics and Differential Equations, vol.
Part 3 presents main results, formulating a nondegeneracy condition on a KAM procedure to guarantee finding tori with fixed frequency, and proving a KAM theorem
for symplectic deformations and a transformed tori theorem.
Using Theorem 1 we can state the main stability result for an elliptic fixed point, known as the KAM theorem (or Kolmogorov Arnold-Moser theorem); see [10, 18, 19].
The KAM theorem says that, under the addition of the remainder term, most of these invariant circles will survive as invariant closed curves under the full map [17, 18].
In order to apply the KAM theorem we have to show that [[gamma].sub.1] [not equal to] 0.
The KAM Theorem leads one to expect that for systems where the interactions among the molecules are non-singular, the phase space will contain islands of stability where the flow is non-ergodic.(11) Thus all of the debate about how ergodicity would explain why phase averages work is purely academic since most of the systems In question are not ergodic, or so all of the available evidence suggests.
There is in fact a theorem, known as the KAM theorem, which says
The infinite real number field is mapped on many different finite floating-point number sets and these sets are not themselves fields (see the KAM theorem