Question about Fractional iteration

Hi.

I remembered this post:
http://groups.google.com/group/sci.math/msg/08fb5b65af84e688?hl=en&dmode=source

Quote:
"
Suppose there is exactly one 2-cycle for f, i.e. exactly one pair of
distinct
points p,q such that f(p) = q and f(q) = p. Then it's easy to prove
there
is no such g. An example is f(x) = -x^3.
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
"

(Here, the "g" referred to is one that solves g(g(x)) = f(x).)

I thought of some additional questions:

1. Does this mean that the map f(x) = x^2 - 1 does not have any half-
function f^(1/2)(x)? Hmm.

2. Is there an analogous theorem for g(g(g(x))) = f(x), perhaps that
if f(x) has exactly one 3-cycle
then g(x) does not exist? That for g^n(x) = f(x), then if f(x) has
exactly one n-cycle, g(x) does not
exist?

3. What about "multivalued" complex "functions" g for complex x?
.

Loading