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Meromorphic functions

  1. Apr 4, 2007 #1
    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.
  2. jcsd
  3. Apr 4, 2007 #2

    matt grime

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    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).

    My first thoughts are that meromorphic functions have Laurent expansions.
  4. Apr 4, 2007 #3


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    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?
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