This is almost an answer, but not quite:

It is known that determining if $(0,0)$ is a point in the attractor of an IFS,
see [this paper][1]. This is very close to what you have.

There is also a very concrete problem, where one iterates a rational map,
and it is unknown if starting with $x=2$ is unbounded. 
I think the map was $x \mapsto x-1/x$ but I might be mistaken.

There are also non-computable julia sets obtained from polynomial iteration,
[see here][2]. 


  [1]: https://www.complex-systems.com/pdf/07-6-2.pdf
  [2]: http://www.math.toronto.edu/yampol/preprints/STOC07.pdf