Pointwise densities of homogeneous Cantor measure and critical values,Nonlinearity

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Pointwise densities of homogeneous Cantor measure and critical values
Nonlinearity ( IF 1.6 ) Pub Date : 2021-04-22 , DOI: 10.1088/1361-6544/abbc62
Derong Kong ₁ , Wenxia Li ₂ , Yuanyuan Yao ₃

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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 system $\left\{{f}_{i}\left(x\right)=\rho x+\frac{i\left(1-\rho \right)}{N-1}:i=0,1,\dots ,N-1\right\}.$ Let s = dim_H E be the Hausdorff dimension of E, and let $\mu ={\mathcal{H}}^{s}{\vert }_{E}$ be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density ${{\Theta}}_{{\ast}}^{s}\left(\mu ,x\right)$ and upper s-density Θ^∗s(μ, x) of μ at x. This extends some early results of Feng etal (2000 J. Math. Anal. Appl. 250 692–705). Furthermore, we determine two critical values a _c and b _c for the sets ${E}_{{\ast}}\left(a\right)=\left\{x\in E:{{\Theta}}_{{\ast}}^{s}\left(\mu ,x\right){\geqslant}a\right\}\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {E}^{{\ast}}\left(b\right)=\left\{x\in E:{{\Theta}}^{{\ast}s}\left(\mu ,x\right){\leqslant}b\right\}$ respectively, such that dim_H E _*(a) > 0 if and only if a < a _c, and that dim_H E*(b) > 0 if and only if b > b _c. We emphasize that both values a _c and b _c are related to the Thue–Morse type sequences, and our strategy to find them relies on ideas from open dynamical systems and techniques from combinatorics on words.

中文翻译：

齐次康托测度和临界值的点密度

让Ñ ⩾2和ρ＆Element;（0,1 / Ñ ² ]。将均匀Cantor集Ë是通过迭代函数系统所产生的自相似组 $\left\{{f}_{i}\left(x\right)=\rho x+\frac{i\left(1-\rho \right)}{N-1}:i=0,1, \dots ,N-1\right\}.$ 设小号=暗淡_ħ Ë是的Hausdorff尺寸ë，让成为s维 Hausdorff 测度仅限于E。在本文中，我们描述了，对于每个x ∈ E，点方向较低的s密度和较高的s密度 Θ ^∗^s ( μ , x ) $\mu ={\mathcal{H}}^{s}{\vert }_{E}$ ${{\Theta}}_{{\ast}}^{s}\left(\mu ,x\right)$ μ在x 处。这扩展了 Feng等人(2000 J. Math. Anal. Appl. 250 692–705) 的一些早期结果。此外，我们确定两个临界值一个 _Ç和b _Ç为套 ${E}_{{\ast}}\left(a\right)=\left\{x\in E:{{\Theta}}_{{\ast}}^{s}\left(\mu ,x\right){\geqslant}a\right\}\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {E}^{{\ast}}\left(b\right) =\left\{x\in E:{{\Theta}}^{{\ast}s}\left(\mu ,x\right){\leqslant}b\right\}$ 分别使得暗淡_ħ Ë _*（一）> 0，当且仅当一个<一个_Ç，而暗淡_ħ Ë *（b）> 0，当且仅当b > b _c。我们强调a _c和b _{c 的}值与 Thue-Morse 类型序列有关，我们找到它们的策略依赖于来自开放动力系统的思想和来自单词组合学的技术。

更新日期：2021-04-22

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