Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE ORBIT OF A β-TRANSFORMATION CANNOT LIE IN A SMALL INTERVAL

  • Kwon, Do-Yong (Department of Mathematics Chonnam National University)
  • Received : 2011.07.23
  • Published : 2012.07.01
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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 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.

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Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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