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Fatou proved a very interesting result: for a transcendental entire function $f$, the second itarate $f^{2}$ has at least has one fixed point. (Using the technique of Picard theorem)

This result has been generalized by Rosenbloom: $f^{n}$ has at least has one fixed point for every $n\geq 2$ (Using the technique of Nevanlinna theory)

Bergweiler proved a very strong version: $f$ has infinitely many points of minimal period $n$, for each $n\geq 2$ (quite technical)

There may be little hope to expect a elementary proof for the general version.

I wonder whether there exist a elementary proof (without using Nevanlinna theory) for the following theorem: $f$ has at least one point of minimal period $2$.

Any reference and comments will be appreciated.

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I edited your question. Do you agree with my changes? – Alexandre Eremenko Sep 8 '13 at 18:04
@yaoxiao Could you clarify if Picard's Theorem is allowed, so that Fatou's result may be applied to reduce the question to the assertion that the equation $$f\circ f(z)-z=e^{g(z)}(f(z)-z)$$ has no entire solutions unless $f$ is the identity? – Adam Epstein Sep 8 '13 at 21:35
@ Alexandre Eremenko, Dear professor, Thanks for your changes of the previous version of the question. – yaoxiao Sep 8 '13 at 23:55

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