Letform a– cycle of an analytic functionThen

The derivative oftakes the same value at each point of the– cycle so tha
Proof
Sincewhere the functionis applied p times, we can deduce using the Chain Rule that
Puttinggives 1 above.
is the product of the derivatives ofat the points of the – cycle sois also the product of the derivatives ofat the points of the– cycle and similarly for the other points
The derivative ofat the point of a– cycle determines the nature of the point. Part 2 above implies that all the points of a cycle have the same nature and we can talk of the nature of the– cycle as a whole.
Ifis a periodic point with periodof an analytic functionthenand the corresponding– cycle are
attracting if
repelling if
indifferent if
super attracting if
Example: Ifthen the period 2 – points areand
sand
soand the 2 – cycle consisting of the pointsandis repelling.