# Talk:Symplectic maps

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