ZETA FUNCTIONS. NO.7

YOSHIFUMI TSUCHIMOTO

Let f be a self map f : M → M of a set M. It defines a (discrete) dynamical system (M, f).

To explain the basic idea, we first examine the case where M is a finite set.

We put A = C(M, ℂ), the set of ℂ-valued functions on M.

f defines a pull-back of functions:

f^{∗}(a)(x) = a(f(x)) (a ∈ A)

and push-forward:

f_{∗}(a)(x) = ∑_{f(y)=x} a(y) (a ∈ A).

(It might be better to treat the push-forward as above as a push-forward of measures.)

We note also that any element of A admits an integration

∫_{M} a = ∑_{x∈M} a(x) (a ∈ A)

(which is a integration with respect to the counting measure.)

Proposition 7.1. We have

∫_{M}(f^{∗}a)b = ∫_{M} a(f_{∗}b)

In other words, f_{∗} is the adjoint of f^{∗}.

Proposition 7.2. Let us put M = {1, 2, …, n}. Let e_{1}, …, e_{n} be the indicators of elements of M. Then {e_{1}, … e_{n}} forms a basis of A. f^{∗} is represented by a matrix P_{f} = (δ_{f(i)j}). f_{∗} is represented by a matrix ^{t}P_{f} = (δ_{if(j)}).

Definition 7.3. We define the set Fix(f) as the set of fixed points of f. Namely,

Fix(f) = {x ∈ M; f(x) = x}.

Proposition 7.4. tr(f^{∗}) = tr(f_{∗}) = # Fix(f).

YOSHIFUMI TSUCHIMOTO

It should be noted that tr((f^{k})^{∗}) may be comuted using a “pathintegral”-like formula.

tr((f^{k})^{∗}) = ∑_{α∈M^{k}} P_{α_{1}α_{2}}P_{α_{2}α_{3}} … P_{α_{k−1}α_{k}}P_{α_{k}α_{1}}

Definition 7.5. We define the Artin-Mazur zeta function of a dynamical system (M, f) as

Z((M, f), T) = exp(∑_{j=1}^{∞} \frac{# Fix(f^{j})T^{j}}{j})

Proposition 7.6.

Z((M, f), T) = \frac{1}{det(1 − T f^{∗})}

7.1. 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 7.7. Let M be a set. Let f : M → M be a map such that # 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(∑_{j=1}^{∞} \frac{# Fix(f^{j})T^{j}}{j})

Let q be a power of a prime p. We may consider an automorphism Frob_{q} of 𝔽̄_{q} over 𝔽_{q} by

Frob_{q}(x) = x^{q}

Proposition 7.8. Frob_{q} : 𝔽_{q^{r}} → 𝔽_{q^{r}} is an automorphism of order r. It is a generator of the Galois group Gal(𝔽_{q^{r}}/𝔽_{q}).

For any projective variety X defined over 𝔽_{q}, we may define a Frobenius action Frob_{q} on X(𝔽̄_{q}):

Frob_{q}([x_{0} : x_{1} : … x_{N}]) = ([x_{0}^{q} : x_{1}^{q} : … x_{N}^{q}]).

For any 𝔽̄_{q}-valued point x ∈ X(𝔽̄_{q}), We have

Frob_{q}^{r}(x) = x ⟺ x ∈ X(𝔽_{q^{r}}).

Proposition 7.9. The Artin Mazur zeta function of the dynamical system (X(𝔽̄_{q}), Frob_{q}) conincides with the congruent zeta function Z(X/𝔽_{q}, t).