Consider a decreasing sequence of positive real numbers $\{\epsilon_n\}_n$ converging to $0$ and a sequence of complex numbers $\{a_n\}_n$, in modulus strictly increasing and diverging to $\infty$.
Define $$ g_n(z)=\frac{\epsilon_n}{z-a_n}\;; $$ $g_n$ is the Möbius transformation corresponding to the matrix $$G_n= \left [ \begin{matrix} 0 & \epsilon_n \\ 1 & a_n \\ \end{matrix} \right ]\;. $$ Is there any way to find a closed form for the composition $h_n=g_n\circ g_{n-1}\circ\cdots\circ g_1$ (respectively the product of matrices $H_n=G_nG_{n-1}\cdots G_1$)?
In particular, fixed $R>|a_n|$, I am interested in finding the 2-dim Lebesgue measure of $$ \{z\in\Bbb C\;:\;|h_n(z)|>R\}\;. $$