A good paper. I enjoyed reading it. I've made small changes in the text. Here are several other comments which may be, will be interesting to the author. Thanks for your good work! It is very helpful. I made corrections which I think adress your concerns 1 and 2 below. I also re-corrected the sentence about the det of a linear symplectic map: Because of Equation (2), a symplectic linear map preserves volume and orientation. Hence the map has determinant 1 and it is invertible. (the det can't be -1)
1. In the section "Linear Symplectic Forms" the sentence "Note that ..." is not clear and probably, literally, not true. I propose to remove it.
I corrected a typo. I think that the statement is true now - I think the intuition of a symplectic form as a sum of 2 X 2 det of projections is important
2. In the section "Types of periodic orbits" the statement "in the 2 dimensional case, an orbit minimizes W if and only if it is hyperbolic " is not true. The following weaker form is ok: "in the 2 dimensional case, any orbit minimizing W is hyperbolic " Thanks! that was indeed a mistake!
3. I would like also to attract attention of the author to a recent book
Treschev D. and Zubelevich O. Introduction to the perturbation theory of Hamiltonian systems. Springer, 2009.
where various aspects of symplectic dynamics are discussed, including KAM-theory, chaos, Arnold diffusion, anti-integrable limit, Hill's formula for periodic solutions, etc. Thanks! I will make sure our library gets it :)
Best regards, Dmitry Treschev