Get ready Part 20

For any real number $\alpha$ ,define the function $f_\alpha(x) =\lfloor x\alpha \rfloor$ . Let $n$ be a positive integer. Does there exists an $\alpha$ such that for $1 \le k \le n,f_{\alpha}^k(n^2)=n^2-k=f_{(\alpha)^k}(n^2)$ ?