Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem

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• 作者：Shigeki Akiyama ; Jonathan Caalim
• 关键词：Beta expansion ; Tarski’s plank problem ; Invariant measures
• 刊名：Discrete & Computational Geometry
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：57
• 期：2
• 页码：357-370
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer US
• ISSN：1432-0444
• 卷排序：57

文摘

We study the invariant measures of a piecewise expanding map in \(\mathbb {R}^m\) defined by an expanding similitude modulo a lattice. Using the result of Bang (Proc Am Math Soc 2:990–993, 1951) on the plank problem of Tarski, we show that when the similarity ratio is at least \(m+1\), the map has an absolutely continuous invariant measure equivalent to the m-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case \(m=2\), we obtain an alternative proof of the result in Akiyama and Caalim (J Math Soc Japan 69:1–19, 2016) together with some improvement.