Author:
ALLOUCHE JEAN-PAUL,CLARKE MATTHEW,SIDOROV NIKITA
Abstract
AbstractLetβ∈(1,2). Eachx∈[0,1/(β−1)] can be represented in the formwhere εk∈{0,1} for allk(aβ-expansion ofx). If$\beta >{(1+\sqrt 5)}/{2}$, then, as is well known, there always existx∈(0,1/(β−1)) which have a uniqueβ-expansion. We study (purely) periodic uniqueβ-expansions and show that for eachn≥2 there exists$\beta _n\in [{(1+\sqrt 5)}/{2},2)$such that there are no unique periodicβ-expansions of smallest periodnforβ≤βnand at least one such expansion forβ>βn. Furthermore, we prove thatβk<βmif and only ifkis less thanmin 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
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