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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: kallecccj@math.leidenuniv.nl)
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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 

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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