Possible Values for Preimage Count in Meromorphic Functions on Riemann Sphere?

[Ant farm]

Hi there,
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.

 

[matt grime]

Hi there,
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...

f^2=f does not imply f=f^-1. Firstly, f^-1 need not exist, indeed cannot exist, unless f=Id. There are also more maps than just Id that satisfy f=f^-1 (or f^2=Id).

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.

My first thoughts are that meromorphic functions have Laurent expansions.

 

[mathwonk]

it seems you have proved that f(f) = f implies degf = 1 or 0.that does sound as if f is id or constant, can you prove that?