Abstract

A self-similar set is a fixed point of iterated function system (IFS) whose maps are similarities. We say that a self-similar set satisfies the common point property if the intersection of images of the attractor under the maps of the IFS is a singleton and this point has a common pre-image, under the maps of the IFS, and the pre-image is in the attractor. Self-similar sets satisfying the common point property were introduced in Sirvent (2008) in the context of space-filling curves. In the present article we study some basic topological and dynamical properties of self-similar sets satisfying the common point property. We show examples of this family of sets. We consider attractors of a sub-IFS, an IFS formed from the original IFS by removing some maps. We put conditions on this attractors for having the common point property, when the original IFS have this property. (C) 2014 Elsevier Ltd. All rights reserved.