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# 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
.*

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2013-06-06