Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T03:21:50.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Isomorphisms between positive and negative $\beta $-transformations

Published online by Cambridge University Press:  09 November 2012

CHARLENE KALLE
CHARLENE KALLE*
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We compare a piecewise linear map with constant slope $\beta \gt 1$ and a piecewise linear map with constant slope $-\beta $. These maps are called the positive and negative $\beta $-transformations. We show that for a certain set of $\beta $s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of $\beta $between 1 and 2 the two maps cannot be isomorphic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 153 - 170
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AHK08]Aihara, K., Hironaka, S. and Kohda, T.. Negative beta encoder. CoRR (2008). Available at: http://arXiv.org/abs/0808.2548.Google Scholar
[AMT97]Ashley, J., Marcus, B. and Tuncel, S.. The classification of one-sided Markov chains. Ergod. Th. & Dynam. Sys. 17(2) (1997), 269–295.Google Scholar
[BB85]Byers, W. and Boyarsky, A.. Absolutely continuous invariant measures that are maximal. Trans. Amer. Math. Soc. 290(1) (1985), 303–314.Google Scholar
[CL00]Cowen, R. and Lungu, E. M.. When are two Markov chains the same? Quaest. Math. 23(4) (2000), 507–513.Google Scholar
[DK11]Dajani, K. and Kalle, C.. Transformations generating negative $\beta $-expansions. Integers 11B (2011), 1–18.Google Scholar
[DMP11]Dombek, D., Masáková, Z. and Pelantová, E.. Number representation using generalized $ (-\beta )$-transformations. Theoret. Comput. Sci. 412 (2011), 6653–6665.Google Scholar
[FL09]Frougny, C. and Lai, A. C.. On negative bases. Proceedings of DLT 09 (Lecture Notes in Computer Science, 5583). Springer, Berlin, 2009, pp. 252–263.Google Scholar
[G{ó}r09]Góra, P.. Invariant densities for piecewise linear maps of the unit interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583.Google Scholar
[HK82]Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1) (1982), 119–140.Google Scholar
[Hof81]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II. Israel J. Math. 38(1–2) (1981), 107–115.Google Scholar
[Hof12]Hofbauer, F.. A two parameter family of piecewise linear transformations with negative slope. Acta Math. Univ. Comenian. 81(1) (2012), 15–30.Google Scholar
[HR02]Hoffman, C. and Rudolph, D.. Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156(1) (2002), 79–101.CrossRefGoogle Scholar
[IS09]Ito, S. and Sadahiro, T.. Beta-expansions with negative bases. Integers 9(A22) (2009), 239–259.Google Scholar
[LS12]Liao, L. and Steiner, W.. Dynamical properties of the negative beta transformation. Ergod. Th. & Dynam. Sys. 32(5) (2012), 1673–1690.Google Scholar
[LY73]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
[LY78]Li, T. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.Google Scholar
[MP11]Masáková, Z. and Pelantová, E.. Ito–Sadahiro numbers vs. Parry numbers. Acta Polytechnica 51 (2011), 59–64.Google Scholar
[NS12]Nakano, F. and Sadahiro, T.. A $({-}\beta )$-expansion associated to Sturmian sequences. Integers 12A (2012), 1–25.Google Scholar
[Orn70]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.Google Scholar
[Par60]Parry, W.. On the $\beta $-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
[R{é}n57]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.Google Scholar