On continuous images of self-similar sets

2020/05/13 by Li, Yuanyuan, Fan, Jiaqi, Gu, Jiangwen +2 · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2005.06163

Abstract

Let (M, c_k, n_k,κ) be a class of homogeneous Moran sets. Suppose f(x,y)∈ C³ is a function defined on ℝ². Given E₁, E₂∈(M, c_k, n_k,κ) , in this paper, we prove, under some checkable conditions on the partial derivatives of f(x,y), that f(E₁,E₂)=\f(x,y):x∈ E₁,y∈ E₂\ is exactly a closed interval or a union of finitely many closed intervals. Similar results for the homogeneous self-similar sets with arbitrary overlaps can be obtained. Further generalization is available for some inhomogeneous self-similar sets if we utilize the approximation theorem.

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