Conjugate Iteration Sequences

The functionsandare conjugate to each other iffor some functioncalled the conjugating function. If the sequenceis defined by

for someandfor

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.