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When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar:

Let f:P^1 -> P^1 be rational function.

Can you say when the sequence {f^n(x)}_n = {x, f(x), f(f(x)), ...} converges? What about the sequence of averages {1/(n+1) \sum_i=0^n f^i(x)}_n? Anything else?