Consider the function
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
The pointis a periodic point, with periodof a functionif:
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.