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I'm curious about what happens if you iterate a function that is periodic. What happens to the period? For example, consider iterating a function like $\sin(x)$ or $\tan(x)$ several times. It should still have period $2\pi$, but will that be its prime period, or can the period become shorter with repeated iteration?

Andrea
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2 Answers2

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It shouldn't take too much imagination to write down a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that

  • The range of $f$ is contained in $[0,1]$
  • $f$ is constant on $[0,1]$
  • The prime period of $f$ is $2$.

What happens to $f \circ f$ in this case?

Mike F
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I don't think the general case is interesting, but I try it for sin and tan. For sin, since sin x lies in [-pi/2 , pi/2], in which sin is monotonic, we have 2pi to be the prime period. For tan, consider tantanx, if T is a period, then tantanT=0,tanT = kpi. For the same reason tan2T=m pi. While tan2T=2tanT/(1-tan^2(T), tanT =0 is the only one possible, and thus T = pi.

Sky
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