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.


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

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