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Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets : a hundred decimal digits for the dimension of E2
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Pollicott, Mark and Jenkinson, Oliver (2018) Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets : a hundred decimal digits for the dimension of E2. Advances in Mathematics, 325 . pp. 87-115. doi:10.1016/j.aim.2017.11.028 ISSN 0001-8708.
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Official URL: https://doi.org/10.1016/j.aim.2017.11.028
Abstract
We prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25.
The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.
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): | Hausdorff compactifications , Continued fractions , Cantor sets, Transfer operators , Determinants | ||||||||
Journal or Publication Title: | Advances in Mathematics | ||||||||
Publisher: | Academic Press | ||||||||
ISSN: | 0001-8708 | ||||||||
Official Date: | 5 February 2018 | ||||||||
Dates: |
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Volume: | 325 | ||||||||
Page Range: | pp. 87-115 | ||||||||
DOI: | 10.1016/j.aim.2017.11.028 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 5 December 2017 | ||||||||
Date of first compliant Open Access: | 6 March 2018 | ||||||||
Open Access Version: |
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