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. 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.

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, is undecidable, see this paper. 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.

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. 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.

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. 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.