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

Abstract

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 $1/\beta$, 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.

17+2 pages, reference updated, Addendum added

References in corpus (1)

• A devil's staircase from rotations and irrationality measures for Liouville numbers