论文标题
Cantor Measure的密度光谱
Density spectrum of Cantor measure
论文作者
论文摘要
给定的$ρ\在（0，1/3] $中，令$μ$为满足$μ= \ frac {1} {2}μf_0^{ - 1}+\ frac {1} {1} {1} {2} {2} {2}μf_1^{ - 1}} $ f_i（$ f_i（$ f_i（$ f_i（$ f_i（$ f_i））， $μ$的支持是迭代功能系统$ \ {f_0，f_1 \} $生成的$ c $。 0} \ frac {μ（b（x，r））} {（2r）^s}，\ qquadθ^{*s}}（μ，x）= \ limsup_ {r \ to 0} \ to 0} \ frac {μ $ C $的Hausdorff尺寸，我们对$ d _*$和$ d^*$的完整描述分别由下层和上部密度的所有可能值组成。高度密度。我们的方法包括在$ [0,1）$上制定涉及二倍地图的问题的“二元”版本，我们通过在某个开放动力学系统的熵上使用已知结果来解决。
Given $ρ\in(0, 1/3]$, let $μ$ be the Cantor measure satisfying $μ=\frac{1}{2}μf_0^{-1}+\frac{1}{2}μf_1^{-1}$, 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 \[ Θ_*^s(μ, x)=\liminf_{r\to 0}\frac{μ(B(x,r))}{(2r)^s},\qquad Θ^{*s}(μ, x)=\limsup_{r\to 0}\frac{μ(B(x,r))}{(2r)^s}, \] where $s=-\log 2/\logρ$ 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.
