length of orbits and fixed point

Given a dynamical system $\frac{dx}{dt}= F(x(t))$

Then is there a relationship between the Cardinal of the fixed point of the classical system $|\operatorname{Fix}(f^{m})|$ with $f^{m}(x)= f(f(\cdots(f(x))$ ($m$ times)

and the length of the orbits of the dynamical system?

For example, I have read the 'Lefschetz fixed point theorem' and it looks to me quite familiar to the explicit formula involving the Chebyshev function

• In 'Lefschetz trace formula' you find $|\operatorname{Fix}(f^{m}(x)|$.

• In 'explicit formula' you find $\frac{d\psi (x)}{dx}$ but this can be understood as a sum over the length of closed orbits (with repetition) $\log p$

-
I don't really understand the question. What is the relation between $f$ and $F$? – Qiaochu Yuan Dec 19 '11 at 21:25