Does $x\_0=1/3$ lead to periodicity in the logistic map $x\_{k+1}=4x\_k(1-x\_k)$?


I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am wondering if there are any known results. After all, $1/3$ is the most rudimentary seed that (I suspect) leads to non-periodicity in the fully chaotic logistic map. Or maybe not?



Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
The orbit of $1/3$ is infinite. You can show this via the [$3$-adic valuation $\nu\_3$](https://en.wikipedia.org/wiki/P-adic_valuation).


Let us show by induction that $\nu\_3(x\_n) = -2^{n}$:
We have $\nu\_3(x\_0) = \nu\_3(1/3) = -1 = -2^0$.


Now $\nu\_3(x\_{n+1}) = \nu\_3(4 x\_{n} (1 - x\_{n})) =
\nu\_3(4)+ \nu\_3(x\_n) + \nu\_3(1 - x\_n)$.


We have $\nu\_3(4) = 0$, and since $\nu\_3(x\_n)$ is negative by induction, we have
$\nu\_3(1 - x\_n) = \nu\_3(x\_n)$, so overall $\nu\_3(x\_{n+1}) = 2\nu\_3(x\_n) =-2^{n+1}$.


From this is is clear that the orbit must be infinite.


A similar argument shows that $p/q$ has infinite orbit for $p, q$ coprime, $q$ divisible by some odd prime.


