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 ↦ 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.
Highlights
Introduction and MotivationThe pioneering work of Rényi [1] and Parry [2] on β-shifts and expansions have motivated a wealth of results providing practical solutions to a variety of problems
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 Tβ,α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers
We examine the structure of the set of eventually periodic points of Tβ,α when β is a Pisot number and when β is the n-th root of a Pisot number
Summary
The pioneering work of Rényi [1] and Parry [2] on β-shifts and expansions have motivated a wealth of results providing practical solutions to a variety of problems. The β-shift Ω β,α is a subshift of finite type if and only if the left shift of the kneading invariants are periodic, see Theorem 4 due to Ito and Takahashi [18], and Parry [19], for the case α ∈ {0, 2 − β}, and Li et al [20], for the case that α ∈ (0, 2 − β) These results immediately give us that the set of parameters in ∆ which give rise to β-shifts of finite type is countable. Kalle and Steiner [25] proved that a β-shift Ω β,α is sofic if and only if its kneading invariants are eventually periodic Combining this result with those of Li et al [5], one obtains that the set of ( β, α) ∈ ∆ with Ω β,α sofic is dense in ∆. We obtain a partial converse and in Corollary 1, we relate these results back to Question A
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