We define G-symmetric polygonal systems of similarities and study the properties of symmetric dendrites, which appear as their attractors. This allows us to find the conditions under which the attractor of a zipper becomes a dendrite.

No abstract

Let S = S 1 , ..., S m be a system of contracting similarities of R 2 . The attractor K(S) of the system S is a non-empty compact set satisfying K = S 1 (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.

We consider fractal squares and obtain the conditions under which they possess finite intersection property. If the fractal square is a dendrite we find the exact estimates for the intersection number for pairs of their pieces and the orders of their points.