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 $M$ to be a finite set.

DEFINITION 07.7   Let $M$ be a set. Let $f: M\to M$ be a map such that $\char93 \operatorname{Fix}(f^n)$ is finite for any $n>0$ . We define the Artin-Mazur zeta function of a dynamical system $(M,f)$ as

$$Z((M,f),T)= \exp(\sum_{j=1}^\infty \frac{\char93 \operatorname{Fix}(f^j) T^j}{j})$$

Let $q$ be a power of a prime $p$ . We may consider an automorphism $\operatorname{Frob}_q$ of $\bar{\mathbb{F}_{q}}$ over $\mathbb{F}_q$ by

$$\operatorname{Frob}_q(x)=x^q$$

PROPOSITION 07.8   $\operatorname{Frob}_q: \mathbb{F}_{q^r}\to \mathbb{F}_{q^r}$ is an automorphism of order $r$ . It is a generator of the Galois group $\operatorname{Gal}(\mathbb{F}_{q^r}/\mathbb{F}_q)$ .

For any projective variety $X$ defined over $\mathbb{F}_q$ , we may define a Frobenius action $\operatorname{Frob}_q$ on $X(\bar{\mathbb{F}_q})$ :

$$\operatorname{Frob}_q([x_0:x_1:\dots x_N]) = ([x_0^q:x_1^q:\dots x_N^q]).$$

For any $\bar{\mathbb{F}_q}$ -valued point $x \in X(\bar{\mathbb{F}_q})$ , We have

$$\operatorname{Frob}_q^r(x)=x \iff x \in X(\mathbb{F}_{q^r}).$$

PROPOSITION 07.9   The Artin Mazur zeta function of the dynamical system $(X(\bar{\mathbb{F}_q}),\operatorname{Frob}_q)$ conincides with the congruent zeta function $Z(X/\mathbb{F}_q,t)$ .