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Discrete & Computational Geometry

, Volume 57, Issue 2, pp 357–370 | Cite as

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

Article
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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.

Keywords

Beta expansion Tarski’s plank problem Invariant measures 

Notes

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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Mathematics & Center for Integrated Research in Fundamental Science and EngineeringUniversity of TsukubaTsukubaJapan
  2. 2.Institute of MathematicsUniversity of the Philippines DilimanQuezon CityPhilippines

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