论文标题
在自相似集的连续图像上
On continuous images of self-similar sets
论文作者
论文摘要
令$（\ Mathcal {M}，C_K，N_K，κ）$为一类同质的Moran集。假设$ f（x，y）\在c^3 $中是在$ \ mathbb {r}^2 $上定义的函数。给定$ e_1，e_2 \ in（\ nathcal {m}，c_k，n_k，κ）$，在本文中，我们在某些可检查条件下证明了$ f（x，y）$，$ f（x，y）$，$ f（e_1，e_1，e_1，e_2）= \ ^ f（x，x，x，x，x，x，x，y y_1，e_1，e_1，间隔或有限的许多封闭间隔的结合。可以获得具有任意重叠的均匀自相似集的类似结果。如果我们利用近似定理，则可以为某些不均匀的自相似集合提供进一步的概括。
Let $(\mathcal{M}, c_k, n_k,κ)$ be a class of homogeneous Moran sets. Suppose $f(x,y)\in C^3$ is a function defined on $\mathbb{R}^2$. Given $E_1, E_2\in(\mathcal{M}, c_k, n_k,κ) $, in this paper, we prove, under some checkable conditions on the partial derivatives of $f(x,y)$, that $$f(E_1,E_2)=\{f(x,y):x\in E_1,y\in E_2\}$$ 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.
