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

TL;DR

This paper characterizes the possible orbit closures of $eta$-transformations, showing they cannot be contained in arbitrarily small intervals, and explicitly determines the maximal domains and minimal intervals for such orbits.

Contribution

It explicitly determines the maximal domain of the function $ ext{ extbackslash Xi}$ and characterizes all minimal intervals containing $T_eta$-orbits, advancing understanding of orbit structures in $eta$-transformations.

Findings

Orbit closures cannot lie in intervals smaller than $1/\beta$

Explicit description of the maximal domain of $ ext{ extbackslash Xi}$

All minimal intervals containing $T_\beta$-orbits are characterized

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.