@GerryMyerson as an example let the matrix $A$ be symplectic and the product be the usual symplectic product

I deeply thank you!

Thank you @WillieWong , I think yours is the only possible explanation. Just to verify if I've understood correctly...why your Taylor expansion lacks the linear term, i.e. why it is $\xi t \to \xi t + O(\xi^2t^2)$ and not $\xi t \to \xi t + O(\xi t)$?

I do prefer your demonstration...unfortunately, I've to follow the one present in the book :-(

I just followed the description given by Arnold, but in fact he treats $V$ as an ordinary parallepiped, and $V'$ as a curvilinear one