Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Abstract

Given $\rho \in (0,1/4]$, the four corner Cantor set $E\subset R^2$ is a self-similar set generated by the iterated function system $\{(\rho x,\rho y),(\rho x,\rho y+1-\rho ),(\rho x+1-\rho ,\rho y),(\rho x+1-\rho ,\rho y+1-\rho )\}.$ For $\theta \in [0,\pi )$ let $E_{\theta }$ be the orthogonal projection of $E$ onto a line with an angle $\theta$ to the $x$-axis. In principle, $E_{\theta }$ is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection $E_{\theta }$ is totally self-similar. We also study the spectrum of $E_{\theta }$, which turns out that the spectrum achieves its maximum value if and only if $E_{\theta }$ is totally self-similar. Furthermore, when $E_{\theta }$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_{\theta }$ which consists of all $x\in E_{\theta }$ having a unique coding. In particular, we show that ${dim}_H{U}_{\theta }={dim}_H{E}_{\theta }$ for Lebesgue almost every $\theta \in [0,\pi )$. Finally, for $\rho =1/4$ we prove that the possibility for $E_{\theta }$ to contain an interval is strictly smaller than that for $E_{\theta }$ to have an exact overlap.