Periodic structure and Schrodinger operators for codings of circle rotations

Abstract

We consider, for any irrational $α$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, α)}$ induced by coding orbits under repeated rotation by $α$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, α)}$ has an associated Schrödinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $α$ has digits with limsup at least $4$, then almost every $c \in X^{(I, α)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(α,I)$ except for a countable set of pairs $(α, |I|)$ where $α$ is Möbius equivalent to the silver mean and $|I| \in \mathbb{Z}α+ f(α)$ where $f(α)$ is a specific infinite series taking value either $\frac{1}{2}, \fracα{2}$, or $\frac{α+1}{2}$. We also show that for a set of $α$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.