Consider the function
and
so that
.and the iterates cycle endlessly between 0 and -1. 0 and -1 are periodic points of f of period 2. They are also fixed points of
Definition
The point
is a periodic point, with period
of a function
if:
but
for
The
points
form a cycle of periodor a– cycle of
Applyingrepeatedly to points of the– cycle just returns other points of the– cycle. Points of the- cycle must be distinct since if
where
then the point
would have to be among the terms
which do not include
which is a contradiction.
All the periodic points ofobviously lie in the keep set
Example: Find the periodic points of period 2 of the function
so we solve
and
are fixed points of f and not period 2 points.
The period 2 points are solutions of
These are
and
Note that
and vice versa.