Let $\Delta_{1}, \Delta_{2}$ be two disjoint closed discs in the Riemann sphere with bounding circles $\Gamma_{1}, \Gamma_{2}$ respectively. Let $J_{k}$ be inversion in the circle $\Gamma_{k}$ and let $T$ be the Möbius transformation $J_{2} \circ J_{1}$.

Show that, if $w \notin \Delta_{1}$, then $T(w) \in \Delta_{2}$ and so $T^{n}(w) \in \Delta_{2}$ for $n=1,2,3, \ldots$ Deduce that $T$ has a fixed point in $\Delta_{2}$ and a second in $\Delta_{1}$.

Deduce that there is a Möbius transformation $A$ with

$$A\left(\Delta_{1}\right)=\{z:|z| \leqslant 1\} \quad \text { and } \quad A\left(\Delta_{2}\right)=\{z:|z| \geqslant R\}$$

for some $R>1$.