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

For $\beta >1$, let $T_{\beta }:[0,1]\to [0,1)$ be the $\beta$-transformation. We consider an invariant $T_{\beta }$-orbit closure contained in a closed interval with diameter $1\mathrm{/}\beta$, then define a function $\mathrm{\Xi }(\alpha ,\beta )$ 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 $\mathrm{\Xi }$, and explicitly specifies all possible minimal intervals containing $T_{\beta }$-orbits. For Addendum: The paper mentioned in the title is completed by this Addendum.