Topology and separation of self-similar fractals in the plane

Abstract

Even though the open set condition (OSC) is generally accepted as the right condition to control overlaps of self-similar sets, it seems unclear how it relates to the actual size of the overlap. For connected self-similar sets in the plane, we prove that a finite overlap implies OSC. On the other hand, there are Cantor sets with arbitrary small dimensions which do not fulfil the OSC.