2022/11/21 by Derong Kong, Beibei Sun, Kong, Derong +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2211.11241
openalex publication_date 2022/11/21 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
Given ρ∈ (0,1/4], the four corner Cantor set E⊂ ℝ2 is a self-similar set generated by the iterated function system \(ρx, ρy), (ρx, ρy+1-ρ), (ρx+1-ρ, ρy), (ρx+1-ρ,ρy+1-ρ)\. For θ∈[0,π) let Eθ be the orthogonal projection of E onto a line with an angle θ to the x-axis. In this paper we give a complete characterization on which the projection Eθ is totally self-similar. We also study the spectrum of Eθ, which turns out that the spectrum of Eθ achieves its maximum value if and only if Eθ is totally self-similar. Furthermore, when Eθ is totally self-similar, we calculate its Hausdorff dimension and study the subset Uθ which consists of all x∈ Eθ having a unique coding. In particular, we show that dimH Uθ=dimH Eθ for Lebesgue almost every θ∈[0,π). Finally, for ρ=1/4 we describe the distribution of θ in which Eθ contains an interval. It turns out that the possibility for Eθ to contain an interval is smaller than that for Eθ to have an exact overlap.