# Zeta functions. No.7

Yoshifumi Tsuchimoto

Congruent zeta as a zeta of a dynamical system.

The definition of Artin Mazur zeta function is valid without assuming the number of the base space to be a finite set.

DEFINITION 7.1   Let be a set. Let be a map such that is finite for any . We define the Artin-Mazur zeta function of a dynamical system as

Let be a power of a prime . We may consider an automorphism of over by

PROPOSITION 7.2   is an automorphism of order . It is a generator of the Galois group .

For any projective variety defined over , we may define a Frobenius action on :

For any -valued point , We have

PROPOSITION 7.3   The Artin Mazur zeta function of the dynamical system conincides with the congruent zeta function .