Pointwise densities of homogeneous Cantor measure and critical values

Let $N\ge 2$ and $\rho\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=\rho x+\frac{i(1-\rho)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $\mu=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $\Theta_*^s(\mu,x)$ and upper $s$-density $\Theta^{*s}(\mu, x)$ of $\mu$ 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 \[ E_*(a)=\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}\quad\textrm{and}\quad E^*(b)=\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\} \] respectively, such that $\dim_H E_*(a)>0$ if and only if $a 0$ 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.