I agree, it's very cool. It features prominently in the study of the modular group (and therefore all things continued fraction). On Thu, Feb 11, 2016 at 5:42 AM, David Wilson <davidwwilson@comcast.net> wrote:
For 2 x 2 matrix M define
F(M) = f : R->R : x => (M11 x + M12) / (M21 x + M22).
Then
F(AB) = F(A) o F(B)
So the composition of unreduced order 1 rational functions is isomorphic to the product of 2x2 matrices.
I assume this is well known, but I thought it was pretty cool.
This means that finding, say, all order 1 rational functions f with
f(f(x)) = x
would reduce to finding all 2x2 matrices M with
M^2 = I.
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