Density spectrum of Cantor measure

Abstract

Given $\rho\in(0, 1/3]$, let $\mu$ be the Cantor measure satisfying $\mu=\frac{1}{2}\mu f_0^{-1}+\frac{1}{2}\mu f_1^{-1}$, where $f_i(x)=\rho x+i(1-\rho)$ for $i=0, 1$. The support of $\mu$ 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 \[ \Theta_*^s(\mu, x)=\liminf_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s},\qquad \Theta^{*s}(\mu, x)=\limsup_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s}, \] where $s=-\log 2/\log\rho$ 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.