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:
(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.)
In other words, is the adjoint of .