We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-\'etale topos satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in characteristic $p$ are well defined on the Weil-\'etale topos. This analogy extends to arithmetic schemes. Over a prime number $p$ and over the archimedean place of $\mathbb{ Q}$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-\'etale topos. This morphism is compatible with the structure mentioned above.
Categories: math.NT math.AG math.DS