EXAMPLES OF USING BINARY CANTORrnSETS TO STUDY THE CONNECTIVITYrnOF SIERPI ´NSKI RELATIVES

摘要

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.