## Tohoku Mathematical Journal 2011 September SECOND SERIES VOL. 63, NO. 3

 Tohoku Math. J. 63 (2011), 329-361

Title SUR L'ANALOGIE ENTRE LE SYSTÈME DYNAMIQUE DE DENINGER ET LE TOPOS WEIL-ÉTALE

Author Baptiste Morin

(Received June 2, 2010, revised July 27, 2010)
Abstract. 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-étale 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-étale topos. This analogy extends to arithmetic schemes. Over a prime number \$p\$ and over the archimedean place of \$\boldsymbol{Q}\$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible with the structure mentioned above.

2000 Mathematics Subject Classification. Primary 14F20; Secondary 14G10, 11R42.

Key words and phrases. Deninger's dynamical system, Weil-étale cohomology, topos.