conformal map disc with two removed points

Question

I need to find all the bijective conformal maps from $D = \{ |z| < 1, z\neq \pm 1/2 \}$ onto itself.

Since this set is not simply connected, I think that the $180°$ degree rotation is the only non-trivial conformal map, but I am not completely sure.

Answer 1

By Riemann's theorem on removable singularities any such map extends to a holomorphic map of unit disk $\Delta$ into $\Delta$. The extended map is still injective: indeed, if a holomorphic map attains some value $w_0$ at $k\ge 1$ points, it attains all neighboring values at least $k$ times. (This follows from Rouché theorem, or from the argument principle.)

This, the extended map is an automorphism of $\Delta$, i.e., a function of the form $$f(z) = \gamma\frac{z-a}{1-\bar az},\qquad |a|<1,\ |\gamma|=1$$ Of these, only the identity map and $f(z)=-z$ fix the set $\{\pm 1/2\}$. (A geometric way to see this is to use the fact that $f$ is an isometry in the hyperbolic metric; in particular, it must send the geodesic $[-1/2,1/2]$ onto itself isometrically. Hence $f(0)=0$, etc.)

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Answer is based on the comment by Greg Martin.