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Monatshefte für Mathematik

, Volume 162, Issue 4, pp 409–427 | Cite as

Simultaneously non-convergent frequencies of words in different expansions

  • David FärmEmail author
Article

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.

Keywords

Interval map Non-typical point Hausdorff dimension Beta shift 

Mathematics Subject Classification (2000)

37E05 37C45 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

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