Your set $A$ might be a parabolic component of the Fatou set of $\mathcal{G}$. In that case $x$ would be an indifferent fixed point of $\mathcal{G}$ on the border of $A$ and for all $y$ in $A$, $\mathcal{G}^n(y)$ would converge to $x$.
The book Iteration of Rational Functions, by Alan F. Beardon, might be useful to you. Beardon focuses specifically on discrete dynamical systems arising from iterating rational functions on the Riemann sphere, but a significant portion of the book focuses on classifying the fixed points of those systems and the properties of the basins of attraction of the fixed points. Perhaps you will find something there to apply to your case.