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I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I am posting it here.

I am looking for references on the uniqueness of the Birkhoff Normal Form (as a formal series) of a Hamiltonian, in a neighborhood of an invariant "KAM torus".

I am also interested in the KAM theory for exact symplectomorphisms, more precisely, I want to know if there is a proof of this kind of theorem using generalized implicit function theorems. Actually any reference in KAM theory in this discrete setting would be appreciated as the bibliography I have found in this topic is quite limited.

Thanks in advance.

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"I want to know if there is a proof of this kind of theorem using generalized implicit function theorems."

Yes, it does exist, see Section 7 of Zehnder, E. Generalized implicit function theorems with applications to some small divisor problems. II. Comm. Pure Appl. Math. 29 (1976), no. 1, 49--111. MR0426055

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