Simultaneously non-convergent frequencies of words in different expansions
(2011) In Monatshefte für Mathematik 162(4). p.409-427- Abstract
- We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. We show that the dimension is preserved when such sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full... (More)
- We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. We show that the dimension is preserved when such sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full dimension even after countable intersections. We also prove that this holds for a dense set of non-integer base expansions. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1925204
- author
- Färm, David LU
- organization
- publishing date
- 2011
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Interval map, Non-typical point, Hausdorff dimension, Beta shift
- in
- Monatshefte für Mathematik
- volume
- 162
- issue
- 4
- pages
- 409 - 427
- publisher
- Springer
- external identifiers
-
- wos:000288804700002
- scopus:79952947922
- ISSN
- 0026-9255
- DOI
- 10.1007/s00605-009-0183-2
- language
- English
- LU publication?
- yes
- id
- 42afd002-5a82-4581-bf7c-9991b7348a98 (old id 1925204)
- date added to LUP
- 2016-04-01 10:59:33
- date last changed
- 2022-03-20 02:00:08
@article{42afd002-5a82-4581-bf7c-9991b7348a98,
abstract = {{We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. We show that the dimension is preserved when such sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full dimension even after countable intersections. We also prove that this holds for a dense set of non-integer base expansions.}},
author = {{Färm, David}},
issn = {{0026-9255}},
keywords = {{Interval map; Non-typical point; Hausdorff dimension; Beta shift}},
language = {{eng}},
number = {{4}},
pages = {{409--427}},
publisher = {{Springer}},
series = {{Monatshefte für Mathematik}},
title = {{Simultaneously non-convergent frequencies of words in different expansions}},
url = {{http://dx.doi.org/10.1007/s00605-009-0183-2}},
doi = {{10.1007/s00605-009-0183-2}},
volume = {{162}},
year = {{2011}},
}