Matching for generalised <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml80" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>β</mml:mi></mml:math>-transformations

Bruin, Henk; Carminati, Carlo; Kalle, Charlene

doi:10.1016/j.indag.2016.11.005

Cited by 10 publications

(22 citation statements)

References 25 publications

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“…Our next proposition (Proposition 1) plays a key rôle in the proof of Theorem 1. We note that after the writing of this paper we became aware of [38] in which a proof of this result also appears. However, for completeness we include a short justification for which we require an auxiliary lemma (Lemma 1).…”

Section: Kalle and Steinermentioning

confidence: 73%

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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: Kalle and Steinermentioning

confidence: 73%

“…Thus, it would be interesting to investigate if there exists other values of β ∈ (1, 2), for which τ + β,α (p) is periodic if and only if τ − β,α (p). In fact this idea is very closely linked to the concept of matching which has recently attracted much attention, see for instance [38,39].…”

Section: Intermediate β-Shifts and Expansionsmentioning

confidence: 99%

Periodic Intermediate β-Expansions of Pisot Numbers

2020

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“…The property that there is some η ∈ N such that f η (c − ) = f η (c + ) is called matching in the literature and it is a subject of current research (see for instance [5], [6]). Combining the first part of the proof of Theorem 4.14 with the above terminology gives the following.…”

Section: Now We Will Prove Thatmentioning

confidence: 99%

“…In the [5], the authors raise the question about parameters β for which generalized β-transformation has matching (this class contain all linear Lorenz maps). They ask whether there exist generalised β-transformations with non-Pisot and non-Salem number β and matching (see end of p.72 in [5]). Our Example 3.1 answers this question affirmatively, showing that such map exists, since map constructed there has matching by Corollary 4.15.…”

Section: Now We Will Prove Thatmentioning

confidence: 99%

“…Finally, in [5], the authors raise the question about parameters β for which generalized β-transformation has matching (this class contain all linear Lorenz maps). They ask whether there exist generalized β-transformations with non-Pisot and non-Salem number β and matching (see the end of p.72 in [5]). Our Example 3.1 answers this question affirmatively, showing that such map exists.…”

Section: Introductionmentioning

confidence: 99%

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Renormalization in Lorenz maps -- completely invariant sets and periodic orbits

Cholewa¹,

Oprocha²

2021

Preprint

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The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. We study connections between periodic points, completely invariant sets and renormalizations. We show that in general, renormalization cannot be fully characterized by a completely invariant set, however there are various situations when such characterization is possible. This way we provide a better insight into the structure of renormalizations in Lorenz maps, correcting some gaps existing in the literature and completing to some extent the description of possible dynamics in this important field of study.

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Natural extensions for p-adic β-shifts and other scaling maps

Furno

2019

Indagationes Mathematicae

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Matching for generalised β-transformations

Cited by 10 publications

References 25 publications

Periodic Intermediate β-Expansions of Pisot Numbers

Periodic Intermediate β-Expansions of Pisot Numbers

Renormalization in Lorenz maps -- completely invariant sets and periodic orbits

Natural extensions for p-adic β-shifts and other scaling maps

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