Isomorphism and bi-Lipschitz equivalence between the univoque sets

In this paper, we consider a class of self-similar sets, denoted by \begin{document}$ \mathcal{A} $\end{document} , and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by \begin{document}$ U_1 $\end{document} . We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of \begin{document}$ U_1 $\end{document} , is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition \begin{document}$ U_1 $\end{document} is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.