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http://dx.doi.org/10.4134/JKMS.2012.49.4.867

THE ORBIT OF A β-TRANSFORMATION CANNOT LIE IN A SMALL INTERVAL  

Kwon, Do-Yong (Department of Mathematics Chonnam National University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.4, 2012 , pp. 867-879 More about this Journal
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.
Keywords
${\beta}$-expansion; ${\beta}$-transformation; Sturmian word; Christoffel word;
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