The orbit of a $\beta$-transformation cannot lie in a small interval

Abstract

For $\beta>1$, let $T_\beta:[0,1]\rightarrow [0,1)$ be the $\beta$-transformation. We consider an invariant $T_\beta$-orbit closure contained in a closed interval with diameter $1/\beta$, then define a function $\Xi(\alpha,\beta)$ by the supremum of such $T_\beta$-orbit with frequency $\alpha$ in base $\beta$, i.e., the maximum value in the $T_\beta$-orbit closure. This paper effectively determines the maximal domain of $\Xi$, and explicitly specifies all possible minimal intervals containing $T_\beta$-orbits. For Addendum: The paper mentioned in the title is completed by this Addendum.