Below are some animations illustrating the behavior of the map
in Problem 2 of the Final Exam:

x_{n+1} =

2x_{n} mod 1

y_{n+1} =

y_{n}/2 if 0 ≤ x_{n} < 1/2
(y_{n}+1)/2 if 1/2 ≤ x_{n} < 1

In each example, the forward or backward iterations of a
region of the square (colored green)
are shown in sequence.
Only the first seven iterations of F or F^{-1}
are shown, and then the sequence repeats.
These are animated GIFs, with a delay of about two seconds between
frames.