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

For $0<\rho <1\mathrm{/}3$ and $\rho \le \lambda \le 1-2\rho$, let $E$ be the self-similar set generated by the iterated function system $$\mathrm{\Phi }=\{{\phi }_1(x)=\rho x,{\phi }_2(x)=\rho x+\lambda ,{\phi }_3(x)=\rho x+1-\rho \}\mathrm{.}$$ All contractive similitudes $f$ with $f(E)\subset E$ are characterized: one can find $i_1,i_2,\dots ,i_n\in \{1,2,3\}$ such that $$f(E)={\phi }_{i_1}\circ {\phi }_{i_2}\circ \cdots \circ {\phi }_{i_n}(E)\mathrm{.}$$