On the pointwise densities of the Cantor measure

Wang, Jun; Wu, Min; Xiong, Yan

doi:10.1016/j.jmaa.2011.01.046

Cited by 6 publications

(6 citation statements)

References 6 publications

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“…In particular, d * = 2d * , independent of ρ. Wang, Wu and Xiong [25] later extended theorem 1.1 to the interval 0 < ρ ρ * , where ρ * ≈ 0.351 811 satisfies (1 − ρ * ) s(ρ * ) = 3/4. However, they showed that for ρ > ρ * , the formula for the upper density changes.…”

Section: Introductionmentioning

confidence: 99%

Density spectrum of Cantor measure

Allaart

Kong

2022

Nonlinearity

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Given ρ ∈ (0, 1/3], let μ be the Cantor measure satisfying μ = 1 2 μ f 0 − 1 + 1 2 μ f 1 − 1 , where f i (x) = ρx + i(1 − ρ) for i = 0, 1. The support of μ is a Cantor set C generated by the iterated function system {f 0, f 1}. Continuing the work of Feng et al (2000) on the pointwise lower and upper densities Θ * s ( μ , x ) = lim inf r → 0 μ ( B ( x , r ) ) ( 2 r ) s , Θ * s ( μ , x ) = lim sup r → 0 μ ( B ( x , r ) ) ( 2 r ) s , where s = −log 2/log ρ is the Hausdorff dimension of C, we give a complete description of the sets D * and D* consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set C. Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our method consists in formulating an equivalent ‘dyadic’ version of the problem involving the doubling map on [0, 1), which we solve by using known results on the entropy of a certain open dynamical system and the notion of tuning.

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Section: Introductionmentioning

confidence: 99%

Density spectrum of Cantor measure

Allaart

Kong

2022

Nonlinearity

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“…The study of densities for a self-similar measure attracted a lot of attention in the literature (see [3,4,7,17,18,19] and the references therein). When N = 2 and ρ ∈ (0, 1/3], Feng et al [8] explicitly calculated the pointwise densities Θ s * (µ, x) and Θ * s (µ, x) for any x ∈ E. The upper bound 1/3 for ρ was later improved to ( √ 3 − 1)/2 by Wang et al [20]. Motivated by their work, Li and Yao [14] determined the pointwise densities of the self-similar measure for non-homogeneous self-similar IFSs.…”

Section: Introductionmentioning

confidence: 99%

Pointwise densities of homogeneous Cantor measure and critical values

Kong,

Li,

Yao

2020

Preprint

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Let N ≥ 2 and ρ ∈ (0, 1/N 2 ]. The homogenous Cantor set E is the self-similar set generated by the iterated function systemLet s = dimH E be the Hausdorff dimension of E, and let µ = H s |E be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density Θ s * (µ, x) and upper s-density Θ * s (µ, x) of µ at x. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values ac and bc for the sets E * (a) = {x ∈ E : Θ s * (µ, x) ≥ a} and E * (b) = {x ∈ E : Θ * s (µ, x) ≤ b} respectively, such that dimH E * (a) > 0 if and only if a < ac, and that dimH E * (b) > 0 if and only if b > bc. We emphasize that both values ac and bc are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.

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“…In particular, d * = 2d * , independent of ρ. Wang et al [20] later extended Theorem 1.1 to the interval 0 < ρ ≤ ρ * , where ρ * ≈ .351811 satisfies (1 − ρ * ) s(ρ * ) = 3/4.…”

Section: Introductionmentioning

confidence: 99%

“…A first observation is that Γ ⊆ C, since lim inf n→∞ T n (x) ∈ C and lim inf n→∞ T n (1−x) ∈ C for every x ∈ C. In [20,Theorem 1.3] Wang et al initiated the study of Γ, but their characterization was rather abstract. By connecting the quantities τ (x) with the theory of unique non-integer base exansions, we are able to give a much more explicit description of Γ here.…”

Section: Introductionmentioning

confidence: 99%

Density spectrum of Cantor measure

Allaart¹,

Kong²

2020

Preprint

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Given ρ ∈ (0, 1/3], let µ be the Cantor measure satisfying µ = 1 2 µf −1 0 + 1 2 µf −1 1 , where fi(x) = ρx + i(1 − ρ) for i = 0, 1. The support of µ is a Cantor set C generated by the iterated function system {f0, f1}. Continuing the work of Feng et al. (2000) on the pointwise lower and upper densitieswhere s = − log 2/ log ρ is the Hausdorff dimension of C, we give a complete description of the sets D * and D * consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set C. Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our proofs are based on recent progress on unique non-integer base expansions and ideas from open dynamical systems.

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On the pointwise densities of the Cantor measure

Cited by 6 publications

References 6 publications

Density spectrum of Cantor measure

Density spectrum of Cantor measure

Pointwise densities of homogeneous Cantor measure and critical values

Density spectrum of Cantor measure

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