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Simultaneously non-convergent frequencies of words in different expansions

Färm, David LU (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:
author
organization
publishing date
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
2011-05-11 13:55:47
date last changed
2017-04-02 03:17:14
@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},
  keyword      = {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},
  volume       = {162},
  year         = {2011},
}