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 pointis a periodic point, with periodof a functionif:

butfor

Thepointsform 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

wherethen the point would have to be among the termswhich do not includewhich 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

andare fixed points of f and not period 2 points.

The period 2 points are solutions ofThese areand

Note thatand vice versa.