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working on some basic questions involving the Riemann Sphere(sigma):C union infinity

firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...

was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...

secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??

stuck here, any hints would be great!

thank you.

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# Meromorphic functions

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