Zeta functions. No.6

Yoshifumi Tsuchimoto

Let $f$ be a self map $f: M\to M$ 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 $A=C(M,\mathbb{C})$ , the set of $\mathbb{C}$ -valued functions on $M$ .

$f$ defines a pull-back of functions:

$$f^*(a)(x)=a(f(x)) \qquad (a\in A)$$

and push-forward:

$$f_*(a)(x)=\sum_{f(y)=x} a(y) \qquad (a\in A).$$

(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

$$\int_M a = \sum_{x \in M} a(x) \qquad (a\in A)$$

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

PROPOSITION 6.1   We have

$$\int_M (f^*a) b = \int_M a (f_* b)$$

In other words, $f_*$ is the adjoint of $f^*$ .

PROPOSITION 6.2   Let us put $M=\{1,2,\dots,n\}$ . Let $e_1,\dots,e_n$ be the indicators of elements of $M$ . Then $\{e_1,\dots e_n\}$ forms a basis of $A$ . $f^*$ is represented by a matrix $P_f=(\delta_{f(i) j})$ . $f_*$ is represented by a matrix ${}^t P_f=(\delta_{i f(j)})$ .

DEFINITION 6.3   We define the set ${\mathrm{Fix}}(f)$ as the set of fixed points of $f$ . Namely,

$${\mathrm{Fix}}(f)=\{x \in M; f(x)=x\}.$$

PROPOSITION 6.4   $\operatorname{tr}(f^*)=\operatorname{tr}(f_*)=\char93 {\mathrm{Fix}}(f).$

It should be noted that $\operatorname{tr}((f^k)^*)$ may be comuted using a ``path-integral''-like formula.

$$\operatorname{tr}((f^k)^*)= \sum_{\alpha\in M^k} P_{\alpha_1\alp... ...} P_{\alpha_2 \alpha_3} \dots P_{\alpha_{k-1} \alpha_k} P_{\alpha_k \alpha_1}$$

DEFINITION 6.5   We define the Artin-Mazur zeta function of a dynamical system $(M,f)$ as

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

PROPOSITION 6.6  

$$Z((M,f),T)= \frac{1}{\det(1-T f^* )}$$