Dendrite-type attractors of IFSs formed by two injective functions

The aim of this paper is to study the dendrite-type attractors of an iterated function system formed by two injective functions. We consider (X, d) a complete metric space and S = (X, {f0, f1}) an iterated function system (IFS), where f0,f1:X⟶X are injective functions and A is the attractor of S. Moreover, we suppose that f0(A)∩f1(A)= {a} and {a}=π(0m1∞)=π(1n0∞) with m, n ≥ 1, where π is the canonical projection on the attractor. We compute the connected components of the sets A\{π(0∞)}, A\{π(1∞)}, A∖{π(0m1∞)=π(1n0∞)} and deduce there are infinitely-many (countably) non-homeomorphic dendrite-type attractors of iterated function systems formed by two injective functions.