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Digit frequencies and self-affine sets with non-empty interior
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Baker, Simon (2018) Digit frequencies and self-affine sets with non-empty interior. Ergodic Theory and Dynamical Systems . doi:10.1017/etds.2018.127 ISSN 0143-3857.
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Official URL: https://doi.org/10.1017/etds.2018.127
Abstract
In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior.
Within expansions in non-integer bases we show that if β∈(1,1.787...) then every x∈(0,1β−1) has a simply normal β-expansion. We also prove that if β∈(1,1+√52) then every x∈(0,1β−1) has a β-expansion for which the digit frequency does not exist, and a β-expansion with limiting frequency of zeros p, where p is any real number sufficiently close to 1/2.
For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to 1, then every non trivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and give rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.
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): | Ergodic theory, Dynamics | ||||||
Journal or Publication Title: | Ergodic Theory and Dynamical Systems | ||||||
Publisher: | Cambridge University Press | ||||||
ISSN: | 0143-3857 | ||||||
Official Date: | 19 December 2018 | ||||||
Dates: |
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DOI: | 10.1017/etds.2018.127 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Reuse Statement (publisher, data, author rights): | This article has been published in a revised form in Ergodic Theory and Dynamical Systems https://doi.org/10.1017/etds.2018.127. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge University Press, 2018 | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Copyright Holders: | Cambridge University Press, 2018 | ||||||
Date of first compliant deposit: | 19 September 2018 | ||||||
Date of first compliant Open Access: | 19 July 2019 | ||||||
RIOXX Funder/Project Grant: |
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