- Let
\[ f (z) = \frac {i - z}{i + z} \quad \mathrm{and} \quad f ^ {- 1} (w) = i \frac {1 - w}{1 + w}. \]
- Given \(\theta \in \mathbb { R }\) , find real numbers \(a , b , c ,\) d such that ad \(- b c = 1\) , and so that for any \(z \in \mathbb { H }\)
\[ \frac {a z + b}{c z + d} = f ^ {- 1} \left(e ^ {i \theta} f (z)\right). \]
- Given \(\alpha \in \mathbb { D }\) find real numbers \(a , b , c ,\) d so that ad \(- b c = 1\) , and so that for any \(z \in \mathbb { H }\)
\[ \frac {a z + b}{c z + d} = f ^ {- 1} \left(\psi_ {\alpha} (f (z))\right), \]
with \(\psi _ { \alpha }\) defined in Section 2.1.
- Prove that if g is an automorphism of the unit disc, then there exist real numbers \(a , b , c ,\) d such that ad \(- b c = 1\) and so that for any \(z \in \mathbb { H }\)
\[ \frac {a z + b}{c z + d} = f ^ {- 1} \circ g \circ f (z). \]
[Hint: Use parts (a) and (b).]
Solution
Goal. Express disc automorphisms, conjugated by the Cayley transform \(f\), as real Möbius transformations of the upper half-plane.
1. \(f(z) = \frac{i-z}{i+z}\) maps \(\HH\) to \(\DD\), with inverse \(f^{-1}(w) = i\frac{1-w}{1+w}\).
the Cayley transform; \(f\) sends the upper half-plane to the unit disk and \(f^{-1}\) is its inverse.
2. (a) For \(\theta \in \RR\), find real \(a,b,c,d\) with \(ad - bc = 1\) and \(\frac{az+b}{cz+d} = f^{-1}(e^{i\theta} f(z))\).
2.1. \(e^{i\theta} f(z)\) is a rotation of the disk, an automorphism of \(\DD\).
multiplication by \(e^{i\theta}\) preserves \(\DD\).
2.2. The composition \(f^{-1} \circ (\text{rotation by } \theta) \circ f\) is a Möbius transformation of \(\HH\) with real coefficients.
\(f\) and \(f^{-1}\) have real coefficients, and the rotation \(w \mapsto e^{i\theta} w\) conjugates to a real Möbius map (the stabilizer of \(i\) in \(\mathrm{PSL}_2(\RR)\)).
2.3. Explicitly, \(f^{-1}(e^{i\theta} f(z)) = \frac{\cos(\theta/2)\, z + \sin(\theta/2)}{-\sin(\theta/2)\, z + \cos(\theta/2)}\).
the rotation by \(\theta\) corresponds, under the Cayley transform, to the matrix \(\begin{pmatrix} \cos(\theta/2) & \sin(\theta/2) \\ -\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}\), which has determinant \(1\) and real entries.
3. (b) For \(\alpha \in \DD\), find real \(a,b,c,d\) with \(ad - bc = 1\) and \(\frac{az+b}{cz+d} = f^{-1}(\psi_\alpha(f(z)))\).
3.1. \(\psi_\alpha(w) = \frac{w - \alpha}{1 - \bar\alpha w}\) is the disc automorphism sending \(\alpha\) to \(0\).
this is the standard automorphism of \(\DD\) (Section 2.1).
3.2. \(f^{-1} \circ \psi_\alpha \circ f\) is a real Möbius transformation of \(\HH\).
it is an automorphism of \(\HH\) (conjugate of an automorphism of \(\DD\)), and every automorphism of \(\HH\) is a real Möbius transformation \(\frac{az+b}{cz+d}\) with \(ad - bc = 1\).
3.3. The coefficients are real and satisfy \(ad - bc = 1\).
\(\mathrm{Aut}(\HH) \cong \mathrm{PSL}_2(\RR)\), realized by matrices of determinant \(1\).
4. (c) Every automorphism \(g\) of \(\DD\) conjugates to a real Möbius transformation of \(\HH\).
4.1. Every \(g \in \mathrm{Aut}(\DD)\) factors as \(g = \psi_\alpha \circ (\text{rotation})\).
\(\mathrm{Aut}(\DD) = \theset{e^{i\theta}\psi_\alpha : \theta \in \RR, \alpha \in \DD}\) (every automorphism is a rotation composed with a \(\psi_\alpha\)).
4.2. Hence \(f^{-1} \circ g \circ f = (f^{-1} \circ \psi_\alpha \circ f) \circ (f^{-1} \circ R_\theta \circ f)\) is a product of real Möbius maps.
by 2 and 3, each factor is a real Möbius transformation.
4.3. The product is a real Möbius transformation \(\frac{az+b}{cz+d}\) with \(ad - bc = 1\).
\(\mathrm{PSL}_2(\RR)\) is closed under composition, and the determinant multiplies to \(1\).
5. Q.E.D.
2, 3, and 4 answer (a), (b), and (c).