Given a recursive homographic sequence $\begin{cases} a_0 \in \mathbb{R} \text{ given} \\ a_{n+1} = f(a_n) \end{cases}$ where $f$ is an homographic function ($x \longmapsto \dfrac{ax+b}{cx+d}$), how can one determine all values of the first term $a_0$ for which the sequence is well defined.
For specific cases, $a_0$ given, one can just proceed by induction to check that for all $n$, $a_n \neq \dfrac{-d}{c}$.
My question is whether there is a general result regarding the well-definiteness of homographic sequences.
Thanks.
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence $\{y_n\}_n$ given by $$y_{n+1}=f^{-1}(y_n)=\frac{dy_n-b}{-cy_n+a} $$ where $y_1=f^{-1}(\infty)=-\frac dc$