Univoque bases and Hausdorff dimension

Kong, Derong; Li, Wenxia; Lü, Fan; Vries, Martijn de

doi:10.1007/s00605-017-1047-9

Cited by 9 publications

(9 citation statements)

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“…Since the differences amongÛ , U L , U R = U and U are at most countable, the dimensional results obtained in this paper for U = U R also hold forÛ , U L and U . Now we recall from [21] the following characterizations of the left and right bifurcation sets U L and U R respectively. where B n (X) denotes the set of all length n subwords occurring in elements of X, and #A denotes the cardinality of a set A.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation sets arising from non-integer base expansions

2019

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Given a positive integer M and q ∈ (1, M +1], let Uq be the set of x ∈ [0, M/(q − 1)] having a unique q-expansion: there exists a unique sequence (xi) = x1x2 . . . with each xi ∈ {0, 1, . . . , M } such thatDenote by Uq the set of corresponding sequences of all points in Uq. It is well-known that the function H : q → h(Uq) is a Devil's staircase, where h(Uq) denotes the topological entropy of Uq. In this paper we give several characterizations of the bifurcation set B := {q ∈ (1, M + 1] : H(p) = H(q) for any p = q} .Note that B is contained in the set U of bases q ∈ (1, M + 1] such that 1 ∈ Uq. By using a transversality technique we also calculate the Hausdorff dimension of the difference U \ B. Interestingly this quantity is always strictly between 0 and 1. When M = 1 the Hausdorff dimension of U \B is log 2 3 log λ * ≈ 0.368699, where λ * is the unique root in (1, 2) of the equation

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Section: Introductionmentioning

confidence: 99%

Bifurcation sets arising from non-integer base expansions

2019

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“…In the sequel, Sidorov [14] used ergodic theoretical methods to prove that for all q ∈ (1,2) almost all x ∈ (0, 1/(1 − q)) have such an expansion. Moreover, Glendinning and Sidorov [7] proved that there always exist (at least countably many) reals having a unique expansion if q > G. Nowadays, especially dimensional theoretical aspects of expansions of reals numbers in non-integer bases are studied; see, for instance, [1,9,10]. In this paper, we introduce non-uniform expansions of real numbers, which may be viewed as expansions with respect to two non-integer bases.…”

Section: Introductionmentioning

confidence: 99%

Non-uniform Expansions of Real Numbers

Neunhäuserer

2021

Mediterr. J. Math.

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“…In [25] the same authors studied the topological properties of U , and showed that its closure U is a Cantor set. Recently, Kong et al proved in [28] that for any q ∈ U we have (1.3) dim H (U ∩ (q − δ, q + δ)) > 0 for any δ > 0.…”

Section: Introductionmentioning

confidence: 99%

On the bifurcation set of unique expansions

Kalle

Kong

et al. 2019

Acta Arith.

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Given a positive integer M , for q ∈ (1, M + 1] let Uq be the set of x ∈ [0, M/(q − 1)] having a unique q-expansion with the digit set {0, 1, . . . , M }, and let Uq be the set of corresponding q-expansions. Recently, Komornik et al. showed in [23] that the topological entropy function H : q → htop(Uq) is a Devil's staircase in (1, M + 1]. Let B be the bifurcation set of H defined by B = {q ∈ (1, M + 1] : H(p) = H(q) for any p = q}.In this paper we analyze the fractal properties of B, and show that for any q ∈ B,where dimH denotes the Hausdorff dimension. Moreover, when q ∈ B the univoque set Uq is dimensionally homogeneous, i.e., dimH (Uq ∩ V ) = dimH Uq for any open set V that intersect Uq.As an application we obtain a dimensional spectrum result for the set U containing all bases q ∈ (1, M + 1] such that 1 admits a unique q-expansion. In particular, we prove that for any t > 1 we have dimH (U ∩ (1, t]) = max q≤t dimH Uq.We also consider the variations of the sets U = U (M ) when M changes.

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Univoque bases and Hausdorff dimension

Abstract: Given a positive integer M and a real number q > 1, a q-expansion of a real number x is a sequence (c i ) = c 1 c 2 . . . with (c i ) ∈ {0, . . . , M} ∞ such that It is well known that if

Cited by 9 publications

References 25 publications

Bifurcation sets arising from non-integer base expansions

Bifurcation sets arising from non-integer base expansions

Non-uniform Expansions of Real Numbers

On the bifurcation set of unique expansions

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