Abstract
Let \(T_{k}\) be the expanding map of [0, 1) defined by \(T_{k}(x) = k x\ \text {mod 1}\), where \(k\ge 2\) is an integer. Given \(0\le a<b\le 1\), let \({\mathcal {W}}_{k}(a,b)=\{x\in [0,1)\ \vert \ T_{k}^nx\notin (a,b), \text { for all } n\ge 0\}\) be the maximal T-invariant subset of \([0,1){\setminus } (a,b)\). We examine the Hausdorff dimension of \({\mathcal {W}}_{k}(a,b)\) as a and b vary.
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The research is partially supported by Center for Research on Environment and Sustainable Technologies (CREST), IISER Bhopal.
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Agarwal, N. The k-Transformation on an Interval with a Hole. Qual. Theory Dyn. Syst. 19, 30 (2020). https://doi.org/10.1007/s12346-020-00383-4
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DOI: https://doi.org/10.1007/s12346-020-00383-4