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Given g:S -> S, the problem of finding f:S -> S such that
f(f(x)) = g(x) for all x in S is a popular one, especially with
S = the real numbers. I have searched all of the internet for research
that has been done on this problem and have found nothing. I have found
recently, about a year after I discovered the following, that my ideas
were not unique. To tell the truth, it scared the sh*t out of me the
first time I searched the entire history of sci.math and found out
that someone had posted my exact ideas long before I had started to
investigate the problem! In addition, I have been told that there is
research on the problem in the lecture "Iterative Function Equations",
the names of the authors I always forget. Consequently this page is
written mostly for the interested amateur. I don't know whether the theory
has been advanced in the direction I have advanced it. Considering I was 15
when I did most of the work, I am happy with it nontheless. If I made any
technical mistakes in the proofs, please let me know; it usually takes
more effort to put a theory in precise mathematical language than
it does in discovering it!
Any contributers will have their results displayed along with
their name.
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