The Library
Entropy, topological transitivity, and dimensional properties of unique q-expansions
Tools
Alcaraz Barrera, Rafael, Baker, Simon and Kong, Derong (2019) Entropy, topological transitivity, and dimensional properties of unique q-expansions. Transactions of the American Mathematical Society, 371 (5). pp. 3209-3259. doi:10.1090/tran/7370 ISSN 0002-9947.
|
PDF
WRAP-entropy-topological-transitivity-unique-expansions-Baker-2019.pdf - Accepted Version - Requires a PDF viewer. Download (872Kb) | Preview |
Official URL: https://doi.org/10.1090/tran/7370
Abstract
Let $ M$ be a positive integer and $ q \in (1,M+1].$ We consider expansions of real numbers in base $ q$ over the alphabet $ \{0,\ldots , M\}$. In particular, we study the set $ \mathcal {U}_{q}$ of real numbers with a unique $ q$-expansion, and the set $ \mathbf {U}_q$ of corresponding sequences.
It was shown by Komornik, Kong, and Li that the function $ H$, which associates to each $ q\in (1, M+1]$ the topological entropy of $ \mathcal {U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $ H$, and characterize the bifurcation set $ \mathscr {E}$ of $ q$'s where the function $ H$ is not locally constant. Moreover, we show that $ \mathscr {E}$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $ (\mathbf {V}_q, \sigma ),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $ \mathcal {U}_q$ coincide for all $ q\in (1,M+1]$.
Item Type: | Journal Article | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Subjects: | Q Science > QC Physics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Entropy | ||||||||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | ||||||||||||
Publisher: | American Mathematical Society | ||||||||||||
ISSN: | 0002-9947 | ||||||||||||
Official Date: | January 2019 | ||||||||||||
Dates: |
|
||||||||||||
Volume: | 371 | ||||||||||||
Number: | 5 | ||||||||||||
Page Range: | pp. 3209-3259 | ||||||||||||
DOI: | 10.1090/tran/7370 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Reuse Statement (publisher, data, author rights): | “First published in Transactions of the American Mathematical Society in 371, 5. 2019 published by the American Mathematical Society,” and the copyright notice in proper form must be placed on all copies. | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Copyright Holders: | American Mathematical Society 2019 | ||||||||||||
Date of first compliant deposit: | 2 October 2017 | ||||||||||||
Date of first compliant Open Access: | 29 January 2019 | ||||||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||||||
Grant number: | EP/M001903/1 | ||||||||||||
RIOXX Funder/Project Grant: |
|
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year