Classificiation Of Fractal Squares’Topological Structure

Given an integer n≥2and a digit set D(?){0,1,…,n-1}2, there is a self-similar set satisfying the set equation:F=(F+D)/n. We call such F a fractal square. We study on the topological structure of fractal squares,and we can classify F into several classes:(i)F is totally disconnected;(ii)the non-trivial components of F are parallel line segments;(iii)F contains a non-trivial component that is not a line segment.Our main idea is that constructing a " fractal torus" H=F+Z2,and putting the question into study on the topological structure of F.Finally,we design a algorithm, that for D={0,…, n-1}2,we could determine F to belong among which classes.Specially,we could determine when F is totally disconnected.