Dynamical systems for arithmetic schemes

We construct a functor from arithmetic schemes (and dominant morphisms) to dynamical systems which allows to recover the Hasse-Weil zeta function of a scheme as a Ruelle type zeta function of the corresponding dynamical system. We state some further properties of this correspondence and explain the relation to the work of Kucharczyk and Scholze who realize the Galois groups of fields containing all roots of unity as (etale) fundamental groups of certain topological spaces. We also explain the main reason why our dynamical systems are not yet the right ones and in what regard they need to be refined.