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 boundary of $A$ and for all $y$ in $A$, $\mathcal{G}^n(y)\to x$.

The book [Iteration of Rational Functions][1], by Alan F. Beardon, might be useful to you.  A significant portion of the book focuses on classifying the fixed points of certain discrete dynamical systems and on the properties of the basins of attraction of those fixed points.  Be aware that Beardon focuses entirely on systems that arise from iterating rational functions on the Riemann sphere, which may or may not apply readily to your case. 

  [1]: http://www.amazon.com/Iteration-Rational-Functions-Dynamical-Mathematics/dp/0387951512