Let $\alpha \in \mathbb{S}^1$ be an algebraic number with $\mathop{\mathrm{arg}}(\alpha)/\pi$ irrational. Is it possible for the rotation by $\alpha$ to converge exponentially fast to a $\xi \in \mathbb{S}^1$? That is, may there exist $\xi \in \mathbb{S}^1$ and $A > 1$ such that $\liminf_{n \to \infty} A^n |\alpha^n - \xi| = 0$?
This answer cites a result of Feldman that implies that if $\xi$ is algebraic and $\xi \neq \alpha^q$ for every $q \in \mathbb{Q}$, then $|n \log \alpha - \log \xi| > C n^{-N}$, where $C > 0$ and $N \geq 1$ are effectively computable numbers depending only on $\alpha$, showing that the rate cannot be exponential in this case.
What about the case where $\xi$ is not algebraic and the case where $\xi = \alpha^q$ for some $q \in \mathbb{Q}$?
Thanks in advance.