In this paper, we consider a class of self-similar sets, denoted by A, 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 U 1. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of U 1 , 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 U 1 is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon. π((i k) ∞ k=1) := lim n→∞ g i1 • • • • • g i k (0). An x ∈ K may have many different codings, if (i k) ∞ k=1 is unique then we call x a univoque point. If x has multiple codings, then usually the attractor K is complicated, see [5, 6]. We say {g j } N j=1 satisfies the open set condition (OSC) [14] if there exists a non-empty bounded open set O ⊆ R such that g i (O) ∩ g j (O) = ∅, i = j
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.