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 $$\Phi =\{{\phi }_1(x)=\rho x,{\phi }_2(x)=\rho x+\lambda ,{\phi }_3(x)=\rho x+1-\rho \}.$$ 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 \[ f(E)=φ_{i_1} \circ φ_{i_2} \circ \cdots \circ φ_{i_n} (E). \]

arXiv comment18 pages