Examples of Using Binary Cantor Sets to Study the Connectivity of Sierpiński Relatives

Abstract

The Sierpiński relatives form a class of fractals that all have the same fractal dimension, but different topologies. This class includes the well-known Sierpiński gasket. Some relatives are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. This paper presents examples of relatives for which binary Cantor sets are relevant for the connectivity. These Cantor sets are variations of the usual middle thirds Cantor set, and their binary descriptions greatly aid in the determination of the connectivity of the corresponding relatives.

Certain Singular Distributions and Fractals

“…"Nowadays, the ternary Cantor set is the paradigmatic model of the fractal geometry [20,45] and in many branches of physics (see [4]). A large class of Cantor-type sets frequently appear as invariant sets and attractors of many dynamical systems of the real world problems, see [17]."…”

Connectivity Properties of Sierpiński Relatives

“…For some relatives, it is possible to use Cantor sets to determine the connectivity properties. 5 Another area for future work is motivated by techniques from shape theory and computational topology. 6,7 One could consider the fractals along with their -hulls as ranges over the non-negative real numbers.…”

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.…”