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Iterating f(x)=2|x|-2

  1. For what initial values of x does the orbit go to infinity?
  2. For what initial values of x is it periodic?
  3. Can you predict the length of the period from the value of the seed?

After some initial experimenting on the calculator, I realized that all seeds between -2 and 2, as entered on the calculator, are periodic. It is not too difficult to prove that in reality, all rational numbers in that range are periodic, while none of the irrational numbers are.

To answer the last question, it becomes necessary to engage in number theoretical thinking, and to support that, I ended up doing some work in Boxer, which can readily be programmed for that sort of computation.

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