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

Abstract

For $0< \rho < 1/3$ and $\rho \le \lambda \le 1-2\rho$, let $E$ be the self-similar set generated by the iterated function system $$\Phi = \big\{ \varphi_1(x) = \rho x ,\; \varphi_2(x) = \rho x + \lambda, \; \varphi_3(x) = \rho x + 1- \rho \big\}.$$ All contractive similitudes $f$ with $f(E) \subset E$ are characterized: one can find $i_1, i_2, \ldots, i_n \in \{1,2,3\}$ such that \[ f(E)=\varphi_{i_1} \circ \varphi_{i_2} \circ \cdots \circ \varphi_{i_n} (E). \]