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Periodic unique beta-expansions: the Sharkovskiĭ ordering

Published online by Cambridge University Press:  01 August 2009

JEAN-PAUL ALLOUCHE,
MATTHEW CLARKE and
NIKITA SIDOROV
JEAN-PAUL ALLOUCHE
Affiliation:
CNRS, LRI, UMR 8623, Université Paris-Sud, Bâtiment 490, F-91405 Orsay Cedex, France (email: allouche@lri.fr)
MATTHEW CLARKE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK (email: M.Clarke@dpmms.cam.ac.uk)
NIKITA SIDOROV
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK (email: sidorov@manchester.ac.uk)
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Abstract

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Let β∈(1,2). Each x∈[0,1/(β−1)] can be represented in the form \[ x=\sum _{k=1}^\infty \varepsilon _k\beta ^{-k}, \] where εk∈{0,1} for all k (a β-expansion of x). If $\beta >{(1+\sqrt 5)}/{2}$, then, as is well known, there always exist x∈(0,1/(β−1)) which have a unique β-expansion. We study (purely) periodic unique β-expansions and show that for each n≥2 there exists $\beta _n\in [{(1+\sqrt 5)}/{2},2)$ such that there are no unique periodic β-expansions of smallest period n for β≤βn and at least one such expansion for β>βn. Furthermore, we prove that βk<βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009, pp. 1055 - 1074
Copyright
Copyright © Cambridge University Press 2009

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