Iterating a periodic function

Question

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?

Answer 1

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?

Answer 2

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.