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On univoque points for self-similar sets

Volume 228 / 2015

Simon Baker, Karma Dajani, Kan Jiang Fundamenta Mathematicae 228 (2015), 265-282 MSC: Primary 37A45; Secondary 37C45. DOI: 10.4064/fm228-3-4

Abstract

Let be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of Daróczy, Kátai, Kallós, Komornik and de Vries.

Authors

  • Simon BakerSchool of Mathematics
    University of Manchester
    Oxford Road
    Manchester, M13 9PL, UK
    e-mail
  • Karma DajaniDepartment of Mathematics
    Utrecht University
    Budapestlaan 6
    3508TA Utrecht, The Netherlands
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  • Kan JiangDepartment of Mathematics
    Utrecht University
    Budapestlaan 6
    3508TA Utrecht, The Netherlands
    e-mail
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