The gap between IFS attractor and strictly self-similar set

An IFS (Iterated Function System) attractor is the unique invariant set of a family of contractions, and a strictly self-similar set is the unique invariant set of a family of similitudes. Any strictly self-similar set is automatically an IFS attractor by definition. For the inverse direction, although the discrepancy between the two concepts is subtle, we herein present a surprising example of an IFS attractor K such that there is no strictly self-similar set possessing the same self-similar topological structure as K .