Let $\Omega=\{z\in \mathbb{C}:1/2 <\Re z<1\}$. Since $\Omega$ is simply connected and not all of $\mathbb{C}$, there is an invertible holomorphic map $g:\Omega\rightarrow \mathbb{D}$, where $\mathbb{D}$ represents the open unit disk in $\mathbb{C}$. Given any biholomorphic self-map $\psi$ of $\Omega$, $g\circ \psi\circ g^{-1}$ is a biholomorphism $\phi$ of $\mathbb{D}$, so $\phi$ must have the form

$\phi(z)=\lambda \displaystyle\frac{z-a}{1-\overline{a}z}$, where $|\lambda|=1$ and $a\in \mathbb{D}$.

Therefore $\psi=g^{-1}\circ\phi\circ g$, so all such $\psi$ must have this form.

While such a map $g$ is not hard to compute, my question is whether there is a "better" classification of biholomorphic self-maps of $\Omega$, perhaps utilizing the symmetry of $\Omega$ using combinations of vertical translations, reflections, etc. I guess what I would like is a nice formula for $\psi$, like those that exist for other "nice" domains (like the upper half-plane, the punctured disk, etc.). My apologies for a somewhat vague question.

Thank you.

  • $\begingroup$ I think your first formulation is best, ’cause it takes into account in the clearest way all the holomorphic automorphisms. $\endgroup$ – Lubin Sep 7 '15 at 2:12

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