I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):


In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there is a theorem that connects the stability of a q-periodic point $(x\_i,y\_i)$ of a symplectomorphism $\psi$ (so we have $(x\_i,y\_i) = \psi(x\_{i-1},y\_{i-1}))$ with the question whether a given sequence of points is a maximizer of minimizer of some discrete action of the form $S =\sum h(x\_i,x\_{i+1}).$ In particular the residue $R = \frac{2- tr(d\psi^{q}(x\_0,y\_0))}{4}$ can be expressed by the Hessian of the action $S$.


I think the equation for the residue was something like $$R = - \frac{det(Hess(S))}{4 \Pi (- \partial\_{1,2} h (x\_i,x\_{i+1}))}.$$


There are clearly a few conditions missing here, but I just hope that anybody here recognizes it. The theorem is quite interesting, as it said more or less something like: The minimizers (stable sequences) of the action, are the dynamically unstable points and vice versa.



Symplectic geometry and stability of orbits
The following paper *Linear stability of periodic orbits in Lagrangian systems* [link](http://www.sciencedirect.com/science/article/pii/0375960183907351) of Mackay and Meiss might be relevant. 


