Discrete & Computational Geometry

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

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

Article

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 

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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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