On deformation of polygonal dendrites preserving the intersection graph

Abstract

Let S = S₁, ..., S_m be a system of contracting similarities of ℝ². The attractor K(S) of the system S is a non-empty compact set satisfying K = S₁(K) ∪ ... ∪ S_m(K). We consider contractible polygonal systems S which are defined by a finite family of polygons whose intersection graph is a tree and therefore the attractor K(S) is a dendrite. We find conditions under which a deformation S′ of a contractible polygonal system S has the same intersection graph and therefore the attractor K(S′) is a self-similar dendrite which is isomorphic to the attractor K of the system S.