There is an efficient geometric tool to visualize functional iteration. The basic idea is simple: Suppose we graph the function $f$ together with the line $y=x\text{.}$ If those two graphs intersect; that point of intersection is a fixed point. Now, suppose we're on the line at the point $(x_i,x_i)\text{.}$ If we move vertically to the graph of the function, we preserve the $x$ coordinate but change the $y$ coordinate to $f(x_i)\text{.}$ Thus, we arrive at the point $(x_i,f(x_i)) = (x_i,x_{i+1})\text{.}$ If we then move horizontally back to the line $y=x$ we now preserve the $y$ coordinate but change the $x$ coordinate so that the $x$ and $y$ coordinates are the same. Thus, we arrive at the point $(x_{i+1},x_{i+1})\text{.}$
In summary: The process of moving vertically from a point on the line $y=x$ to the graph of $f$ and back to the line horizontally is a geometric representation of one application of the function $f\text{.}$ This step is illustrated in Figure 1(a). Repeated application of this process represents repeated application of $f\text{,}$ i.e. iteration. This is illustrated in Figure 1(b). Note that the orbit appears to be attractive.