Next: Congruent zeta as a

# Zeta functions. No.7

Yoshifumi Tsuchimoto

Let be a self map of a set . It defines a (discrete) dynamical system .

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

We put , the set of -valued functions on .

defines a pull-back of functions:

and push-forward:

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

We note also that any element of admits an integration

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

PROPOSITION 7.1   We have

In other words, is the adjoint of .

PROPOSITION 7.2   Let us put . Let be the indicators of elements of . Then forms a basis of . is represented by a matrix . is represented by a matrix .

DEFINITION 7.3   We define the set as the set of fixed points of . Namely,

PROPOSITION 7.4

It should be noted that may be comuted using a path-integral''-like formula.

DEFINITION 7.5   We define the Artin-Mazur zeta function of a dynamical system as

PROPOSITION 7.6

Next: Congruent zeta as a
2015-05-28