Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$, where $f^m$ stands for the $m$-th iterate.
1. Can one describe/classify all such pairs?
This is probably very hard, and perhaps there exists no simple answer. But here is a simpler question:
2. Is there an algorithm which finds out whether two rational functions have a common iterate or not ?
I mean, I give you two rational functions, say with integer coefficients, and you tell me whether they have a common iterate or not. Perhaps using a super-computer...
Motivation. J. F. Ritt, (Permutable rational functions. Trans. Amer. Math. Soc. 25 (1923), no. 3, 399-448) gave a complete classification/description of all commuting pairs of rational functions (that is $f(g)=g(f)$)... except when they have a common iterate. I gave a completely different proof of Ritt's theorem, but again it does not apply to the case when $f$ and $g$ have a common iterate (MR1027462).
Polynomial pairs (commuting, or with a common iterate) are completely described in MR1501149 Ritt, J. F. On the iteration of rational functions. Trans. Amer. Math. Soc. 21 (1920), no. 3, 348-356, in the very end of this paper.
What is the exact relation between permutable pairs and pairs with a common iterate ?
3. If two functions have a common iterate, must they commute?
Or perhaps they must, but with explicitly listed exceptions? A positive answer to this will solve problem 2 above. See also my "answer" to http://mathoverflow.net/questions/48818 for an additional motivation.
EDIT. And one more question:
4. Can one describe commuting functions that have a common iterate?
This would complete Ritt's description of commuting functions.