**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.)