Iterated function systems whose attractors are Cantor sets

n this paper we consider classical iterated function systems (IFS) consisting of a finite number of contracting mappings for a complete metric space. The main goal is to study the class of IFSs whose attractors are Cantor sets, i.e. perfect totally disconnected sets. Important representatives of this class are totally disconnected IFSs introduced by Barnsley.
We have proposed other definitions of a totally disconnected IFS and proved their equivalence to the Barnsley definition. Sufficient conditions for IFS to be totally disconnected are obtained.
It is shown that injectivity of mappings from an IFS implies the perfection of the attractor and its uncountability. Also it is proved that if the mappings from an IFS are injective and the sum of their contraction coefficients is less than one, then the attractor is a Cantor set.
In general case, these conditions do not guarantee totally disconnectedness of an IFS.
Meanwhile, it is shown that if an IFS consists of two injective mappings and the sum of their contraction coefficients is less than one, then the IFS is totally disconnected. Examples of IFS attractors are constructed, demonstrating that conditions of the proven theorems are only sufficient but not necessary.