• Kwon, Do-Yong
  • Received : 2011.07.23
  • Published : 2012.07.01


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 such $T_{\beta}$-orbit with frequency ${\alpha}$ in base ${\beta}$, i.e., the maximum value in $T_{\beta}$-orbit closure. This paper effectively determines the maximal domain of ${\Xi}$, and explicitly specifies all possible minimal intervals containing $T_{\beta}$-orbits.


${\beta}$-expansion;${\beta}$-transformation;Sturmian word;Christoffel word


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Supported by : National Research Foundation of Korea(NRF)