On continuous images of self-similar sets

[摘要] Let (M, c(k), n(k), k) be a class of homogeneous Moran sets. Suppose f (x, y) is an element of C-3 is a function defined on R-2. Given E-1, E-2 is an element of (M, c(k),n(k), k), in this paper, we prove, under some checkable conditions on the partial derivatives of f(x, y), that f(E-1, E-2) - {f(x, y) : x is an element of E-1, y is an element of E-2} 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. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Continuous images;Self-similar sets;Intervals