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.