Let $x = \mathbb{R}^n$ and $f_{i}:\mathbb{R}^n\rightarrow \mathbb{R}$ be polynomials (of coordinates). Let $F :\mathbb{R}^n\rightarrow \mathbb{R}^n = (f_1,\ldots,f_n)$.
For some $x$ and $F$, one can plot the scatter plots of $I(x) = \{ F^{(k)}(x) \}_{k\ge 1}$ (where $F^{(k)}$ mean iterations) and somehow guess that the image looks like a bounded region of certain form. One will then want to prove that $I(x)$ is dense in that region.
EDIT: Someone asked for a (non-trivial) example. Try $n=2$ with
$$f_1(x,y) = y^2-2x-4, f_2(x,y) = x, F=(f_1,f_2), (x_0,y_0)=(3/4,2).$$
The scatter plot will look like
and one can easily prove that the absolute value of each coordinate of the points is less than 4.
Which branch of mathematics systematically studies this kind of question? I kind of recall that people talk about function iterations in dynamical systems, and there is also something called ergodic theory, but I failed to find references that directly connect them with the question above.