Conjugate Iteration Sequences
The functionsandare conjugate to each other iffor some functioncalled the conjugating function. If the sequenceis defined by
then the sequencesatisfiesfor
andandare called conjugate iteration sequences.
In practice the functionis usually found by substitutingandintoand rearranging.
Since the sequenceis the image of the sequenceunder the functionboth sequences must have the same behaviour of convergence and continuity and ifis a fixed point ofthenis a fixed point of
If the conjugating function is to be one to one and entire then it must be of the form
Example: Show that the sequence
is conjugate to the sequencewith conjugating function
Note first thatis one to one onIfthen sobecomes
This simplifies to
so thatandare conjugate functions and the sequencesandare conjugate sequences.