Simultaneously nonconvergent frequencies of words in different expansions
(2011) In Monatshefte für Mathematik 162(4). p.409427 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 noninteger base expansions. (Less)
Please use this url to cite or link to this publication:
http://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, Nontypical 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
 00269255
 DOI
 10.1007/s0060500901832
 language
 English
 LU publication?
 yes
 id
 42afd0025a824581bf7c9991b7348a98 (old id 1925204)
 date added to LUP
 20110511 13:55:47
 date last changed
 20180107 04:41:11
@article{42afd0025a824581bf7c9991b7348a98, 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 noninteger base expansions.}, author = {Färm, David}, issn = {00269255}, keyword = {Interval map,Nontypical point,Hausdorff dimension,Beta shift}, language = {eng}, number = {4}, pages = {409427}, publisher = {Springer}, series = {Monatshefte für Mathematik}, title = {Simultaneously nonconvergent frequencies of words in different expansions}, url = {http://dx.doi.org/10.1007/s0060500901832}, volume = {162}, year = {2011}, }