# Colloquium

### Dynamical systems for arithmetic schemes

We construct a natural infinite dimensional dynamical system whose periodic orbits come in compact packets $P$ which are in bijection with the prime numbers $p$. Here each periodic orbit in $P$ has length $\log p$. In fact a corresponding construction works more generally for finitely generated normal rings and their maximal ideals or even more generally for arithmetic schemes and their closed points. Moreover the construction is functorial for a large class of morphisms. Thus the zeta functions of analytic number theory and arithmetic geometry can be viewed as Ruelle type zeta functions of dynamical systems. We will describe the construction and what is known about these dynamical systems. The generic fibres of our dynamical systems are related to an earlier construction by Robert Kucharczyk and Peter Scholze of topological spaces whose fundamental groups realize Galois groups. There are many unproven conjectures on arithmetic zeta functions and the ultimate aim is to use analytical methods for dynamical systems to prove them.