Self-embeddings of homogeneous self-similar sets generated by three maps

2025/09/20 by Wang, Zhiqiang
#28A80 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.18202

Abstract

For 0< ρ< 1/3 and ρ≤ λ≤ 1-2ρ, let E be the self-similar set generated by the iterated function system Φ= \ φ₁(x) = ρx , φ₂(x) = ρx + λ, φ₃(x) = ρx + 1- ρ\. All contractive similitudes f with f(E) ⊂ E are characterized: one can find i₁, i₂, …, i_n ∈ \1,2,3\ such that f(E)=φ_i1 ∘ φ_i2 ∘ ⋯ ∘ φ_in (E).

Citations

• On a self-embedding problem of self-similar sets
• On a kind of self-similar sets with complete overlaps
• On the structure of λ-Cantor set with overlaps
• Self embeddings of Bedford-McMullen carpets
• Some problems on the boundary of fractal geometry and additive combinatorics
• Self-similar and self-affine sets; measure of the intersection of two copies

Related

• On a self-embedding problem of self-similar sets
• On a kind of self-similar sets with complete overlaps
• A general theory of self-similarity II: recognition
• Self-similar measures associated to a homogeneous system of three maps
• New results on embeddings of self-similar sets via renormalization