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?

  • $\begingroup$ I have never heard of these zeta functions but they seem pretty interesting, where did you come across this? $\endgroup$ – Tony S.F. Apr 4 '16 at 3:15
  • $\begingroup$ there are many zeta functions en.wikipedia.org/wiki/List_of_zeta_functions not all are directly related to the Riemann zeta, but all have some common properties with an other zeta function (so that studying one could be useful to understand them all) and represent some interesting and "non-trivial" informations on the space they are based on. most of them have a meromorphic continuation, some sort of Euler product representing some sort of "primes" on the space, and ideally an equivalent of the Riemann hypothesis $\endgroup$ – reuns Apr 4 '16 at 4:23
  • $\begingroup$ question : do you have some archetypal example of your zeta function ? $\endgroup$ – reuns Apr 4 '16 at 4:28
  • $\begingroup$ @user1952009 The archetypal example for an arithmetical zeta function would be a Riemann zeta function or a Dedekind zeta function. For dynamical zeta functions the archetypal example would be the one above. $\endgroup$ – Mohamed Alaa El Behairy Apr 4 '16 at 15:08
  • $\begingroup$ @MohamedAlaaElBehairy : very funny. I am asking a concrete example, not the definition : a concrete dynamical system, how you compute its fixed points, and the function you get. $\endgroup$ – reuns Apr 5 '16 at 1:30

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).

  • $\begingroup$ That is very interesting, do you know if there is any general framework in which both of these zeta functions are special cases? your example provides an interpretation of one in terms of the other but I think there are other examples of zeta functions that are not related to either yet exhibit the same closely related set of properties $\endgroup$ – Mohamed Alaa El Behairy Apr 4 '16 at 17:29
  • $\begingroup$ @MohamedAlaaElBehairy : finding some interesting common properties of multiplicative and additive zeta functions seems difficult, no ? $\endgroup$ – reuns Apr 6 '16 at 3:26
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    $\begingroup$ what do you mean by a multiplicative zeta function? $\endgroup$ – Mohamed Alaa El Behairy Apr 6 '16 at 3:48

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