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Bifurcation sets arising from non-integer base expansions
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Allaart, Pieter, Baker, Simon and Kong, Derong (2019) Bifurcation sets arising from non-integer base expansions. Journal of Fractal Geometry, 6 (4). pp. 301-341. doi:10.4171/JFG/79 ISSN 2308-1309 .
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Official URL: https://doi.org/10.4171/JFG/79
Abstract
Given a positive integer M and q∈(1,M+1], let Uq be the set of x∈[0,M/(q−1)] having a unique q-expansion: there exists a unique sequence (xi)=x1x2… with each xi∈{0,1,…,M} such that
x=x1q+x2q2+x3q3+⋯.
Denote by Uq the set of corresponding sequences of all points in Uq. It is well-known that the function H:q↦h(Uq) is a Devil's staircase, where h(Uq) denotes the topological entropy of Uq. In this paper we give several characterizations of the bifurcation set
B:={q∈(1,M+1]:H(p)≠H(q) for any p≠q}.
Note that B is contained in the set {UR} of bases q∈(1,M+1] such that 1∈Uq. By using a transversality technique we also calculate the Hausdorff dimension of the difference U∖B. Interestingly this quantity is always strictly between 0 and 1. When M=1 the Hausdorff dimension of U∖B is log23logλ∗≈0.368699, where λ∗ is the unique root in (1,2) of the equation x5−x4−x3−2x2+x+1=0.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Bifurcation theory, Set theory | |||||||||
Journal or Publication Title: | Journal of Fractal Geometry | |||||||||
Publisher: | European Mathematical Society Publishing House | |||||||||
ISSN: | 2308-1309 | |||||||||
Official Date: | 30 September 2019 | |||||||||
Dates: |
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Volume: | 6 | |||||||||
Number: | 4 | |||||||||
Page Range: | pp. 301-341 | |||||||||
DOI: | 10.4171/JFG/79 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Date of first compliant deposit: | 1 February 2018 | |||||||||
Date of first compliant Open Access: | 1 February 2018 | |||||||||
RIOXX Funder/Project Grant: |
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