A result on the approximation properties of the orbit of 1 under the β-transformation

Cao, Chun-Yun

doi:10.1016/j.jmaa.2014.05.049

Cited by 5 publications

(1 citation statement)

References 13 publications

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“…for almost all β > 1 (for the Lebesgue measure), was studied by Persson and Schmeling [148] [174], Ban and Li [16], Cao [49], Fang, Wu and Li [76], Li and Chen [118], Lü and Wu [126], Tan and Wang [196]. Kwon [108] studies the subset of Parry numbers whose conjugates lie close to the unit circle, using technics of combinatorics of words.…”

Section: 3mentioning

confidence: 99%

A proof of the Conjecture of Lehmer

Verger-Gaugry¹

2019

Preprint

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The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f α (z) associated with the dynamical zeta functions ζ α (z) of the Rényi-Parry arithmetical dynamical systems (β -shift), for α a reciprocal algebraic integer α of house α greater than 1, (ii) the discovery of lenticuli of poles of ζ α (z) which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when α tends to 1 + , giving rise to a continuous lenticular minorant M r ( α ) of the Mahler measure M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant M r ( α ) as a function of the dynamical degree. The Conjecture of Schinzel-Zassenhaus is proved to be true. A Dobrowolski type minoration of the Mahler measure M(α) is obtained. The universal minorant of M(α) obtained is θ −1 η > 1, for some integer η ≥ 259, where θ η is the positive real root of −1 + x + x η . The set of Salem numbers is shown to be bounded from below by the Perron number θ −1 31 = 1.08545 . . ., dominant root of the trinomial −1 − z 30 + z 31 . Whether Lehmer's number is the smallest Salem number remains open. For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number Θ = 1.3247 . . ., whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on |z| = 1 (limit equidistribution).

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Section: 3mentioning

confidence: 99%

A proof of the Conjecture of Lehmer

Verger-Gaugry¹

2019

Preprint

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Diophantine analysis in beta-dynamical systems and Hausdorff dimensions

Lü

2016

Advances in Mathematics

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The -transformation with a hole at 0

Kalle¹,

Kong²,

Langeveld³

et al. 2019

Ergod. Th. Dynam. Sys.

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For β ∈ (1, 2] the β-transformation T β : [0, 1) → [0, 1) is defined by T β (x) = βx (mod 1). For t ∈ [0, 1) let K β (t) be the survivor set of T β with hole (0, t) given byfor all n ≥ 0 . In this paper we characterise the bifurcation set E β of all parameters t ∈ [0, 1) for which the set valued function t → K β (t) is not locally constant. We show that E β is a Lebesgue null set of full Hausdorff dimension for all β ∈ (1, 2). We prove that for Lebesgue almost every β ∈ (1, 2) the bifurcation set E β contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of β ∈ (1, 2) for which E β contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for E 2 , the bifurcation set of the doubling map. Finally, we give for each β ∈ (1, 2) a lower and upper bound for the value τ β , such that the Hausdorff dimension of K β (t) is positive if and only if t < τ β . We show that τ β ≤ 1 − 1 β for all β ∈ (1, 2).Urbański proved that the function t → h top (g|K g (t)) is a Devil's staircase, where h top denotes the topological entropy.Motivated by the work of Urbański, we consider this situation for the β-transformation. Given β ∈ (1, 2], the β-transformation T β : [0, 1) → [0, 1) is defined by T β (x) = βx

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A result on the approximation properties of the orbit of 1 under the β-transformation

Cited by 5 publications

References 13 publications

A proof of the Conjecture of Lehmer

A proof of the Conjecture of Lehmer

Diophantine analysis in beta-dynamical systems and Hausdorff dimensions

The -transformation with a hole at 0

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