Prove that a loxodromic transformation can be expressed as a resultant of elliptic and hyperbolic transformations.
Thanks and Regards,
Kalyan.
Hi,
Here is a proof I came up with any bilinear transformation $\displaystyle L(\omega, z)$ with two fixed points say $\displaystyle \alpha, \beta$ will be of the form $\displaystyle \frac{\omega - \alpha}{\omega - \beta} = k.\frac{z - \alpha}{z - \beta} $ when $\displaystyle k$ is neither unimodular nor $\displaystyle k \in R $ the transformation is neither elliptical nor hyperbolic in the respective cases, hence loxodromic.
Now for all $\displaystyle k \in C $ we know that $\displaystyle k = r.(\cos\theta + i\sin\theta) , r \in R$ and $\displaystyle |\cos\theta + i\sin\theta| = 1$
$\displaystyle \therefore$ we can express the transformation $\displaystyle L(\omega, z)$ can be expressed as $\displaystyle E(H(\frac{z - \alpha}{z - \beta}))$ where $\displaystyle E(t) = (cis\theta)t$ and $\displaystyle H(t) = r.t , r \in R$
where $\displaystyle E(t), H(t)$ are elliptic and hyperbolic transformations respectively.
The notation used for $\displaystyle E(t), H(t)$ may be confusing but what I mean is its just a matrix multiplication.
Let me know if this proof is ok.
Kalyan.