Connectivity Properties of Sierpiński Relatives

Abstract

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.

Using Epsilon Hulls to Characterize and Classify Totally Disconnected Sierpiński Relatives

“…This generalizes methods presented in Ref. 2. To define what is meant by a "hole" in an -hull, we consider the boundary of a hole.…”

“…Note that for the relatives, being totally disconnected is equivalent to being totally path-disconnected. 2 The theorem is originally stated and proved in. 2…”

“…Moreover, the conditions for connectivity are as follows. 2 (1) A relative R is totally path-disconnected if and only if one of the following hold:…”

Topological structure of fractal squares

“…We would like to thank Professor Huo-Jun Ruan for inspiring discussions. We are also grateful to the referee for the valuable comments and suggestions, and, in particular, for pointing out the two latest papers of Taylor et al [17,18].…”

“…Subsequently, there are many works devoted to study such topological properties ( [1], [2], [3], [7], [9], [10], [11], [13], [14], [15]). Recently, Taylor et al [17,18] considered the connectedness properties of the Sierpinski relatives with rotations and reflections. Xi and Xiong [19] showed that for a fractal square, it is totally disconnected if and only if the number of cells in the connected components in each iteration is uniformly bounded.…”