  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.
