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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) $\log p$

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

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