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In ChaosBook, at page 61 of the unstable version of the book, it is stated that

$$J_p (x) \mu (x) = \mu (x,)$$ i.e the velocity vector is an eigenvector of the Jacobian along periodic orbit $p$ with eigenvalue 1.

Moreover, in the $3$rd weeks video lectures with titled Types of Floquet multipliers, Dr. Cvitanovic states that

There has to be good reason why $J$ - Jacobian matrix along the periodic orbit - has eigenvalue one; there are two possibilities: One is symmetries, [...]

This is a mind blowing statement, and I would like to know whether there is any physical or mathematical argument why that must be the case. Of course, that might just be an empirical observation, but even in that case, I would like know whether there is at least an intuitive argument for why that might the case.

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Prof. Cvitanovic states that eigenvalue 1 in the monodromy matrix is a non-generic case with Lebesgue measure zero. It should only occur for a reason, such as, symmetry or bifurcation, cf. ChaosBook, Section 5.1.

On the other hand if a Lagrangian system has a symmetry, then there is a conserved quantity $Q$ by Noether's theorem. In other words, the flow of the symmetry and the flow of the time-evolution commute. Therefore if the symmetry flow leads to an infinitesimal variation of the periodic orbit, the infinitesimal variation should be constant along the orbit, corresponding to eigenvalue 1.

For a Hamiltonian system the observation by Prof. Cvitanovic follows from Poincare theorem, cf. my Phys.SE answer here.

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