I have an exercise to suggest.
Let $E_{2020}: \mathbb S^1 \to \mathbb S^1$ such that $E_{2020}(x)=\{2020x\}=2020x \, \text{mod} \, \, 1$.
Find a point $x_0\in [0,1)$ such that $E_{2020}(x)$ is eventually periodic.
A point $x_0$ is eventually periodic if there exist $p\in \mathbb N$ such that $E_{2020}^p$ is periodic.
Given $x\in [0,1)$ we have $x =\displaystyle \sum_{k\geq 1}\frac{a_k}{10^k}$. Then $2020x = \displaystyle\sum_{k\geq 1}\frac{(2\times 10^3 + 2 \times 10)a_k}{10^k}$. Applying the map $E_{2020}$, we get: \begin{align*} E_{2020}= & 2 \left(\displaystyle \sum_{k\geq 4}\frac{a_k}{10^{k-3}} + \displaystyle \sum_{k\geq 2}\frac{a_k}{10^{k-1}} \right)\\ =& 2 \left(\displaystyle \sum_{k\geq 1}\frac{a_{k+3}}{10^k} + \displaystyle \sum_{k\geq 1}\frac{a_{k+1}}{10^k}\right)\\ =& 2 \left(\displaystyle \sum_{k\geq 1}\frac{a_{k+3}+a_{k + 1}}{10^k}\right). \end{align*} So I dare a response: $x=.\epsilon_0 0 \epsilon_2 0 \cdots$ where $\{\epsilon_k\}\subset \{0,1\}^{\mathbb N}$ (decimal writing with odd-ranked zeros).
Any contribution is welcome.