A Note On Topology Of Fractal Squares With Order Three

We consider a family of fractal squares, denoted as F3,7. Each of them satisfies the set equation K = 1/3(K + D) for some D subset of {0, 1, 2}(2) with #D = 7. It is known that two of these fractal squares are Lipschitz equivalent if and only if they are isometrically equivalent. The aim of our study is to improve this by replacing Lipschitz equivalence with topological equivalence. To this end, we shall investigate the group G(aut)(K) of all homeomorphisms of a fractal square K is an element of F-3,F-7 that has a cut point and show that #G(aut)(K) = 2 or 8.