Periodic intermediate β-expansions of Pisot numbers

Blaine Quackenbush, Tony Samuel, Matt West

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Abstract

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β-shifts, namely transformations of the form T β, α: x → βx +α mod 1 acting on [-α/(β-1), (1-α)/(β-1)], where (β, α) ∈ Δ is fixed and where Δ:= {(β, α) ∈ R 2: β ∈ (1, 2) and 0 ≤ α ≤ 2-β}. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of (β, α) such that Tb, a has the subshift of finite type property is dense in the parameter space Δ. Here, they proposed the following question. Given a fixed β ∈ (1, 2) which is the n-th root of a Perron number, does there exists a dense set of a in the fiber {β} x (0, 2-b), so that T β, α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when α = 0 to the case when α ∈ (0, 2-β). That is, we examine the structure of the set of eventually periodic points of T β, α when β is a Pisot number and when β is the n-th root of a Pisot number.

Original languageEnglish
Article number903
JournalMathematics - Special Issue "Fractals: Geometry, Analysis and Mathematical Physics"
Volume8
Issue number6
DOIs
Publication statusPublished - 1 Jun 2020

Bibliographical note

Funding Information:
Funding: Part of the work presented here was carried out during an undergraduate research project supported by the Bill and Linda Frost Fund. The second author would also like to extend thanks to the Hausdorff Research Institute for Mathematics and Institut Mittag Leffler for their kind hospitality where the final writing stages of this article took place.

Publisher Copyright:
© 2020 by the authors.

Keywords

  • β-expansions; shifts of finite type; periodic points; iterated function systems
  • 37E05, 37B10, 11A67, 11R06

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