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

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Abstract

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.

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Acknowledgements

We would like to express our gratitude to Peter Grabner and Wöden Kusner who informed us of the result of Bang, which made Theorem 1.1 in the present form. The first author is indebted to Hiroyuki Tasaki for stimulating discussion. The authors are supported by the Japanese Society for the Promotion of Science (JSPS), Grant in aid 21540012. The second author expresses his deepest gratitude to the Hitachi Scholarship Foundation.

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Authors and Affiliations

  1. Institute of Mathematics & Center for Integrated Research in Fundamental Science and Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 350-8571, Japan

    Shigeki Akiyama & Jonathan Caalim

  2. Institute of Mathematics, University of the Philippines Diliman, 1101, Quezon City, Philippines

    Jonathan Caalim

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  1. Shigeki Akiyama
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  2. Jonathan Caalim
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Correspondence to Shigeki Akiyama.

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Akiyama, S., Caalim, J. Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem. Discrete Comput Geom 57, 357–370 (2017). https://doi.org/10.1007/s00454-016-9849-4

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  • DOI: https://doi.org/10.1007/s00454-016-9849-4

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