Skip to main content Accessibility help

Dynamical properties of the negative beta-transformation


We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

Hide All
[1]Allouche, J.-P., Arnold, A., Berstel, J., Brlek, S., Jockusch, W., Plouffe, S. and Sagan, B. E.. A relative of the Thue–Morse sequence. Discrete Math. 139(1–3) (1995), 455461.
[2]Bertrand, A.. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér. A 285(6) (1977), 419421.
[3]Dubickas, A.. On the distance from a rational power to the nearest integer. J. Number Theory 117(1) (2006), 222239.
[4]Dubickas, A.. On a sequence related to that of Thue–Morse and its applications. Discrete Math. 307(9–10) (2007), 10821093.
[5]Faller, B.. Contribution to the ergodic theory of piecewise monotone continuous maps. PhD Thesis, École Polytechnique Fédérale de Lausanne, 2008.
[6]Frougny, C. and Lai, A. C.. On negative bases. Proceedings of DLT 09 (Lecture Notes in Computer Science, 5583). Springer, Berlin, 2009, pp. 252263.
[7]Góra, P.. Invariant densities for generalized β-maps. Ergod. Th. & Dynam. Sys. 27(5) (2007), 15831598.
[8]Hofbauer, F.. A two parameter family of piecewise linear transformations with negative slope. Acta Math. Univ. Comenian. (N.S.) to appear.
[9]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34(3) (1979), 213237.
[10]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II. Israel J. Math. 38(1–2) (1981), 107115.
[11]Hofbauer, F.. The structure of piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 1(2) (1981), 159178.
[12]Ito, S. and Sadahiro, T.. Beta-expansions with negative bases. Integers 9(A22) (2009), 239259.
[13]Keller, G.. Piecewise monotonic transformations and exactness. Seminar on Probability (Rennes, 1978). Université de Rennes, Rennes, 1978, Exp. No. 6, p. 32 (in French).
[14]Li, T.-Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183192.
[15]Lothaire, M.. Combinatorics on words. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1997.
[16]Masáková, Z. and Pelantová, E.. Ito–Sadahiro numbers vs. Parry numbers, arXiv:1010.6181v1 [math.NT].
[17]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401416.
[18]Pollicott, M. and Yuri, M.. Dynamical Systems and Ergodic Theory (London Mathematical Society Student Texts, 40). Cambridge University Press, Cambridge, 1998.
[19]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477493.
[20]Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499530.
[21]Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4) (1980), 269278.
[22]Wagner, G.. The ergodic behaviour of piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 46(3) (1979), 317324.
[23]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121153.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed