next up previous
Next: Congruent zeta as a


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:

% latex2html id marker 756
$\displaystyle f^*(a)(x)=a(f(x))
\qquad (a\in A)

and push-forward:

% latex2html id marker 758
$\displaystyle 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

% latex2html id marker 762
$\displaystyle \int_M a = \sum_{x \in M} a(x)
\qquad (a\in A)

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

PROPOSITION 07.1   We have

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

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

PROPOSITION 07.2   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 07.3   We define the set $ \operatorname{Fix}(f)$ as the set of fixed points of $ f$ . Namely,

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

PROPOSITION 07.4   $ \operatorname{tr}(f^*)=\operatorname{tr}(f_*)=\char93 \operatorname{Fix}(f).

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

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

DEFINITION 07.5   We define the Artin-Mazur zeta function of a dynamical system $ (M,f)$ as

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


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

next up previous
Next: Congruent zeta as a
Yoshifumi Tsuchimoto 2016-06-18