论文标题
均匀的cantor量度和临界值的密度
Pointwise densities of homogeneous Cantor measure and critical values
论文作者
论文摘要
令$ n \ ge 2 $和$ρ\在（0,1/n^2] $中 \ [ \ left \ {f_i（x）=ρx+\ frac {i（1-ρ）} {n-1}：i = 0,1，\ ldots，n-1 \ right \}。 \] 令$ s = \ dim_h e $为$ e $的Hausdorff尺寸，让$μ= \ Mathcal h^s | _e $是$ s $ s $二维的hausdorff度量，限于$ e $。在本文中，我们描述的是，对于E $中的每个$ X \，下点$ s $ s $ s $θ_*^s（μ，x）$和$ x $ $ x $ $μ$的上限$ s $ s $ s $ s $θ^{*s}（μ，x）$。这扩展了冯等人的一些早期结果。 （2000）。此外，我们确定两个关键值$ a_c $和$ b_c $的集合 \ [ e _*（a）= \ left \ {x \ in E：θ_*^s（μ，x）\ ge a \ right \} \ quad \ textrm {and} \ quad e^*（b）= \ \ lest \ left \ left \ left \ per \] 分别使$ \ dim_h e _*（a）> 0 $ i时，并且仅当$ a <a_c $，而当时$ \ dim_h e^*（b）> 0 $ ifly和仅当$ b> b_c $。我们强调，两个值$ a_c $和$ b_c $都与thue-morse类型序列有关，我们的策略依赖于从单词上的组合动力学中的开放动态和技术中的想法。
Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, 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 $a_c$ and $b_c$ for the sets \[ E_*(a)=\left\{x\in E: Θ_*^s(μ, x)\ge a\right\}\quad\textrm{and}\quad E^*(b)=\left\{x\in E: Θ^{*s}(μ, x)\le b\right\} \] 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 dynamics and techniques from combinatorics on words.
