两个反演之积的不动点

hbghlyj

$L_1,L_2$是与单位圆正交的圆
3、作关于$L_1,L_2$之积的不动点$p,q$？

显然，不动点$p,q$满足：
$p$关于$L_1$的反演点是$q$，$p$关于$L_2$的反演点也是$q$

hbghlyj

我知道了。任取$z$。过$z,T(z),T^2(z)$作圆，与单位圆的交点就是$p,q$。

hbghlyj

Lesson on Hyperbolic Translations
$\displaystyle{ T_\alpha = \frac{z+\alpha}{\bar{\alpha}z +1} , |\alpha|<1 .}$
the two fixed points of translation are the two points, $±α/|α |$, where the diameter through $α$ meets the unit circle. Thus $Tα$ has no fixed hyperbolic points.

hbghlyj

上面是用Poincaré disk model的。那两个不动点在边界圆上。 如果用半平面模型，当然，那两个不动点也是在边界上：people.reed.edu/~ormsbyk/341/SL2R.html

Hyperbolic transformations

When $\mathrm{tr}^2\sigma > 4$, we call $\sigma$ a hyperbolic transformation. Here $\sigma$ has two distinct fixed points on $\mathbb{R} = \partial H^2$ and we think of $\sigma$ as a "hyperbolic translation." Arcs of circles passing through both fixed points are stable under $\sigma$, and geodesics with one of the fixed points as center are taken to each other. The following animation illustrates how $\begin{pmatrix} e^t & 0 \\ e^t-e^{-t} & e^{-t} \end{pmatrix}$ acts on some geodesics with center either $0$ or $1$ as $t$ varies from $0$ to $3$.