Bifurcation sets arising from non-integer base expansions

  • Pieter Allaart

    University of North Texas, Denton, USA
  • Simon Baker

    University of Birmingham, UK
  • Derong Kong

    Chongqing University, China
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Abstract

Given a positive integer and , let be the set of having a unique -expansion: there exists a unique sequence with each such that

Denote by the set of corresponding sequences of all points in . It is well-known that the function is a Devil's staircase, where denotes the topological entropy of . In this paper we give several characterizations of the bifurcation set

Note that is contained in the set {} of bases such that . By using a transversality technique we also calculate the Hausdorff dimension of the difference . Interestingly this quantity is always strictly between 0 and 1. When the Hausdorff dimension of is , where is the unique root in of the equation .

Cite this article

Pieter Allaart, Simon Baker, Derong Kong, Bifurcation sets arising from non-integer base expansions. J. Fractal Geom. 6 (2019), no. 4, pp. 301–341

DOI 10.4171/JFG/79