What is the relationship between all the dynamical zeta functions and the number theoretical zeta functions?

Question

One can associate to any dynamical system a zeta function based on counting the number of fixed points of the iterates of the transformation. Explicitly we have: $$\zeta_{A} = exp \left( \sum_{n=1} \frac{1}{n}\left | Fix(f^{n}) \right | z^{n} \right)$$ One can similarly define other zeta functions associated to dynamical systems of all sorts. These zeta functions admit a product representation, a meromorphic continuation and the other nice properties of zeta functions. My question is : is there any rigorous relationship between these dynamical zeta functions and the arithmetical ones beyond merely analogy or inspiration?

Answer 1

Yes, the relationship comes from looking at the dynamical zeta function of the Frobenius map acting on the $\overline{\mathbb{F}}_p$-points of a variety over $\mathbb{F}_p$. This reproduces the zeta function of the variety in the usual sense (the one the Weil conjectures are about).