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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.

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The following paper Linear stability of periodic orbits in Lagrangian systems link of Mackay and Meiss might be relevant.

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