Give a countable set of periodic points of $T$ where $\Bbb T=[0,1)/\{0=1\}$

Question

Let $T : \Bbb T → \Bbb T$ be the doubling map defined as $T(x) = 2x (\mod 1)$. Give a countable set of periodic points of $T$ where $\Bbb T=[0,1)/\{0=1\}$

I was trying to use the shift map on $Σ^+_2 = \{0, 1\}^{\Bbb N}$, but can't really get it.

Answer 1

For $n ≥ 2$, let $x_n = w_nw_nw_n · · · ∈ Σ^+_2$ , where $w_n$ is a finite word with one $1$ followed by $(n−1) 0$’s. The number in $\Bbb T = [0, 1)$ with binary expansion $x_n$ is $\sum_{k=0}^∞\frac{1}{2^{kn+1}} =\frac{2^{n−1}}{2^{n} − 1}$ . Note that the least period of $x_n$ is $n$.