On a self-embedding problem of self-similar sets

Abstract: Let $K\subset\mathbb{R}^d$ be a self-similar set generated by an iterated function system ${\varphi_i}_{i=1}^m$ satisfying the strong separation condition and let $f$ be a contracting similitude with $f(K)\subset K$. We show that $f(K)$ is relative open in $K$ if all $\varphi_i$'s share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question in Elekes, Keleti and M{\'a}th{\'e} [Ergodic Theory Dynam. Systems 30 (2010)]. As a byproduct of our argument, when $d=1$ and $K$ admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction parts of opposite signs, we show that $K$ is symmetric. This partially answers a question in Feng and Wang [Adv. Math. 222 (2009)].