On univoque Pisot numbers

Allouche, Jean‐Paul; Frougny, Christiane; Hare, Kathryn E.

doi:10.1090/s0025-5718-07-01961-8

Cited by 16 publications

(31 citation statements)

References 27 publications

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“…Each point ω + of the greedy β-shift and each point ω − of the lazy β-shift corresponds to a β-expansion of a unique point in [0, 1/(β − 1)]. Note, if ω + and ω − are β-expansions of the same point, then ω + and ω − are not necessarily equal, see Example 1 and [6,13,14]. There are several ways in which one may generate β-expansions of real numbers, other than using the greedy and lazy β-shifts.…”

Section: Introductionmentioning

confidence: 99%

Periodic Intermediate β-Expansions of Pisot Numbers

2020

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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 ↦ β x + 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 T β , α 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 α in the fiber { β } × ( 0 , 2 − β ) , 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.

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Section: Introductionmentioning

confidence: 99%

Periodic Intermediate β-Expansions of Pisot Numbers

2020

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“…Also note that for any b ≥ 2, the real number β such that the β-expansion of 1 is b1 ∞ is a univoque Pisot number in (b, b + 1). It would be interesting to determine the smallest univoque Pisot number in (b, b+1): the case b = 1 was addressed in [5], but the proof uses heavily the fine structure of Pisot numbers in (1, 2) (see [8,20,21]). A similar study of Pisot numbers in (b, b + 1) would certainly help.…”

Section: Discussionmentioning

confidence: 99%

Univoque numbers and an avatar of Thue–Morse

Allouche

Frougny

2009

Acta Arith.

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Univoque numbers are real numbers λ > 1 such that the number 1 admits a unique expansion in base λ, i.e., a unique expansion 1 = j≥0 a j λ −(j+1) , with a j ∈ {0, 1, . . . , ⌈λ⌉ − 1} for every j ≥ 0. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called admissible sequences. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set {0, 1, . . . , b}, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval (b, b + 1), where b is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism 3 → 31, 2 → 30, 1 → 03, 0 → 02, occurs in a "universal" manner.

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“…From a computational point of view, an algorithm to compute Pisot numbers is known [6,8,9]. A study of Pisot numbers with unique beta-expansion has been started [1,16].…”

Section: Definitionmentioning

confidence: 99%

“…Amara [2] has determined all the limit points of the Pisot numbers smaller than 2. (1,2) are the following:…”

Section: Infinite Families Of Finite Reversibly Greedy Pisot Numbersmentioning

confidence: 99%