On deformation of polygonal dendrites preserving the intersection graph

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.

Elementary fractal geometry. 4. Automata-generated topological spaces

“…There are a lot of recent papers on fractal squares, mainly by Chinese authors. See [15,24,34,42,43,[50][51][52] and their references. The automaton for fractal squares is the same as for the corresponding square, with fewer edge labels.…”

“…So we must ask for characteristic exponents rather than absolute invariants. A first step would be to characterize contractible spaces which include trees [15] and disk-like tiles [1,29,32,49].…”