J. Korean Math. Soc. 2012; 49(4): 867-879
Printed July 1, 2012
https://doi.org/10.4134/JKMS.2012.49.4.867
Copyright © The Korean Mathematical Society.
DoYong Kwon
Chonnam National University
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.
Keywords: $\beta$-expansion, $\beta$-transformation, Sturmian word, Christoffel word
MSC numbers: 11A63, 37B10, 68R15
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