In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be decomposed into small pieces, such that each one admits a so-called generating function of type $V$, which is a kind of analogue of the Hamiltonian function itself, but rather associated with the small piece only, and satisfying a discrete analogue of Hamilton's equations. The discrete action functional is then defined on a space of discrete trajectories by means of this generating function and a discrete analogue of the Liouville form. I refer to p.366 of the book for more details (this is pretty involved in terms of details).
On the other hand, Chaperon has defined a generating function by means of the technique of broken trajectories, to prove the Arnold conjecture for the $2n$-torus in a finite-dimensional technical way. Laudenbach and Sikorav have extended his construction to the case of cotangent bundles (I find their article more readable than Chaperon's). In addition to proving a cup length version of the Arnold conjecture, this allowed them to establish the existence of a generating function quadratic at infinity for any Hamiltonian deformation of the zero section of a cotangent bundle (over a closed manifold).
In Mc Duff and Salamon's book, it is said that Chaperon's generating function and the discrete action functional are the same (p.363). I would like to know if by any chance, someone could help me understand why this is true.
The edition of Mc Duff and Salamon's book I am talking about is the last one, edition 2017.
Thanks a lot in advance.
