I am asking myself a question. Let $\alpha > 1$ and $\{x\}$ denote the fractional part of $x$ which is $x - \lfloor x \rfloor$. Let $\{ u_n(x) =\alpha^n x \}_{n \in \mathbb{N^{*}}}$ Given $x \in [-\infty,0[ \cup ]1, +\infty[$. Could we find $x' \in [0,1]$ such that $\{ u_n(x) =\alpha^n x \}_{n \in \mathbb{N^{*}}}$ and $\{ u_n(x') =\alpha^n x' \}_{n \in \mathbb{N^{*}}}$ contain exactly same elements which mean if we take an element of $u_n(x)$ we can find it in $u_n(x')$.
Thanks in advance !