Criterion for substitutivity of Sturmian palindromes and one-dimensional factor dynamics (Q2027884)

Let \(\alpha\) be some irrational. For any \(n\in\mathbb{Z}\) suppose \(x_n=1\) if \(\{n\alpha\}\in [1-\frac{\alpha}{2};1)\cup [0;\frac{\alpha}{2}]\), and \(x_n=0\) otherwise. The sequence \(\{x_n\}\) can be considered as a biinfinite word over the binary aphabet. It is proved that this word is a morphic palindrome if and only if \(\alpha\) is a quadratic irrational.