The similarity boundary of a self-similar set

Abstract

Abstract. We define the similarity boundary of a self-similar set and use it to analyze theproperties of self-similar sets in the general setting of any complete metric space. The similarityboundary is an attempt at extending the concept of the topological boundary in a way that isconsistent with our intuitive understanding of the term boundary. We also show how with theanalysis of the similarity boundary, we can restrict ourselves to the space K and disregard theambient space. §1. IntroductionWe analyze self-similar sets which arise as the invariant sets or attractors of a finitecollection of similitudes on a complete metric space. Let {f 1 ,··· ,f N } be a collection ofcontracting similitudes with contraction factors {r 1 ,··· ,r N }, defined on a complete metricspace (X,d). Then there exists a unique, non-empty, compact subset K ⊆ X such thatK =S Ni=1 f i (K). This set K, which is the attractor of the collection of similitudes is calledthe self-similar set or the invariant set of the similitudes. The collection of similitudes iscalled an iterated function system (IFS).There are a number of dimensions associated with a self-similar set. Of these, one ofthe most important and widely studied dimensions is the Hausdorff dimension and theassociated Hausdorff measure. In fact, one of the earliest characterizations of self-similarsets with positive Hausdorff measure gave rise to the famous