Discrete & Computational Geometry

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

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



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.


Beta expansion Tarski’s plank problem Invariant measures 



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.


  1. 1.
    Akiyama, S., Caalim, J.: Rotational beta expansion: ergodicity and soficness. J. Math. Soc. Japan 69, 1–19 (2016)MathSciNetGoogle Scholar
  2. 2.
    Akiyama, S., Pethő, A.: On canonical number systems. Theor. Comput. Sci. 270(1–2), 921–933 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Akiyama, S., Scheicher, K.: Symmetric shift radix systems and finite expansions. Math. Pannon 18(1), 101–124 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Ball, K.: The plank problem for symmetric bodies. Invent. Math. 104(3), 535–543 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bang, T.: A solution of the “plank problem”. Proc. Am. Math. Soc. 2, 990–993 (1951)MathSciNetMATHGoogle Scholar
  6. 6.
    Buzzi, J., Keller, G.: Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps. Ergodic Theory Dyn. Syst. 21(3), 689–716 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dajani, K., de Vries, M.: Invariant densities for random \(\beta \)-expansions. J. Eur. Math. Soc. 9(1), 157–176 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gilbert, W.J.: Complex numbers with three radix representations. Can. J. Math. 34, 1335–1348 (1982)CrossRefMATHGoogle Scholar
  9. 9.
    Góra, P.: Invariant densities for generalized \(\beta \)-maps. Ergodic Theory Dyn. Syst. 27(5), 1583–1598 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Góra, P., Boyarsky, A.: Absolutely continuous invariant measures for piecewise expanding \(C^2\) transformation in \({{\bf R}}^N\). Isr. J. Math. 67(3), 272–286 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ito, Sh, Sadahiro, T.: Beta-expansions with negative bases. Integers 9(A22), 239–259 (2009)MathSciNetMATHGoogle Scholar
  12. 12.
    Ito, Sh, Takahashi, Y.: Markov subshifts and realization of \(\beta \)-expansions. J. Math. Soc. Japan 26(1), 33–55 (1974)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kalle, C.: Isomorphisms between positive and negative \(\beta \)-transformations. Ergodic Theory Dyn. Syst. 34(1), 153–170 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kátai, I., Szabó, J.: Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37, 255–260 (1975)MathSciNetMATHGoogle Scholar
  15. 15.
    Keller, G.: Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d’une région bornée du plan, C. C.R. Acad. Sci. Paris Sér. A-B 289(12), A625–A627 (1979)MATHGoogle Scholar
  16. 16.
    Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69(3), 461–478 (1985)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kempton, T.: On the invariant density of the random \(\beta \)-transformation. Acta Math. Hung. 142(2), 403–419 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kenyon, R.: The construction of self-similar tilings. Geom. Funct. Anal. 6, 471–488 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kenyon, R., Solomyak, B.: On the characterization of expansion maps for self-affine tilings. Discrete Comput. Geom. 43(3), 577–593 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Komornik, V., Loreti, P.: Expansions in complex bases. Can. Math. Bull. 50(3), 399–408 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lagarias, J.C., Wang, Y.: Integral self-affine tiles in \({\mathbb{R}}^n\) I. Standard and nonstandard digit sets. J. Lond. Math. Soc. 54(2), 161–179 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, T.Y., Yorke, J.A.: Ergodic transformations from an interval into itself. Trans. Am. Math. Soc. 235, 183–192 (1978)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Liao, L., Steiner, W.: Dynamical properties of the negative beta-transformation. Ergodic Theory Dyn. Syst. 32(5), 1673–1690 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Parry, W.: On the \(\beta \)-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11, 401–416 (1960)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Parry, W.: Representations for real numbers. Acta Math. Acad. Sci. Hung. 15, 95–105 (1964)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Parry, W.: The Lorenz attractor and a related population model, Ergodic theory. In: Proceedings Conference Mathematics Forschungsinstitut Oberwolfach, 1978. Lecture Notes in Mathematics, vol. 729, pp. 169–187. Springer, Berlin (1979)Google Scholar
  27. 27.
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Safer, T.: Polygonal radix representations of complex numbers. Theor. Comput. Sci. 210(1), 159–171 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223–248 (2000)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Scheicher, K., Thuswaldner, J.M.: Canonical number systems, counting automata and fractals. Math. Proc. Camb. Philos. Soc. 133(1), 163–182 (2002)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Tarski, A.: Further remarks about degree of equivalence on polygons (English translation by I. Wirszup), Collected Papers, vol. 1, pp. 597–611. Birkhäuser, Basel, Boston, Stuttgart (1986)Google Scholar
  32. 32.
    Tsujii, M.: Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane. Commun. Math. Phys. 208(3), 605–622 (2000)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tsujii, M.: Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143(2), 349–373 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Vince, A.: Self-replicating tiles and their boundary. Discrete Comput. Geom. 21, 463–476 (1999)MathSciNetCrossRefMATHGoogle Scholar

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© 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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