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Average frequencies of digits in infinite IFS’s and applications to continued fractions and Lüroth expansions

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Date
12/08/2020
Author
Olsen, Lars
West, M.
Keywords
Baire category
Non-normal numbers
Average systems
Infinite iterated function systems
Continued fraction expansion
Lüroth expansion
Hausdorff dimension
Packing dimension
QA Mathematics
T-NDAS
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Abstract
The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form: a typical (in the sense of Baire) point x ∈ K has the following property: the average frequencies of digits of x have maximal oscillation. We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Hölder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point x∈K has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Lüroth expansion digits.
Citation
Olsen , L & West , M 2020 , ' Average frequencies of digits in infinite IFS’s and applications to continued fractions and Lüroth expansions ' , Monatshefte für Mathematik , vol. First Online . https://doi.org/10.1007/s00605-020-01457-w
Publication
Monatshefte für Mathematik
Status
Peer reviewed
DOI
https://doi.org/10.1007/s00605-020-01457-w
ISSN
0026-9255
Type
Journal article
Rights
Copyright © The Author(s) 2020. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/20481

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