Graphical Iteration

Given an initial value x0 and a function f, we generate the sequence
x0, x1 = f(x0), x2 = f(x1), x3 = f(x2), ...
The algebraic mechanism is clear, though perhaps unenlightening.
Graphical iteration is a geometrical method for visualizing this process.
Briefly, we use an alternating collection of vertical and horizontal segments to connect these points.
(x0,0) to (x0,f(x0)) = (x0,x1)
(x0,x1) to (x1,x1)
(x1,x1) to (x1,f(x1)) = (x1,x2)
(x1,x2) to (x2,x2)
and so on.
Here is an illustration of the first few steps of graphical iteration.
Here is an example of sensitivity to initial conditions for the tent map.

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