Simple infinitely ramified self-similar sets

Abstract

Self-similar sets form a mathematically tractable class of fractals. Any family of contractive mappings f1., fm generates a unique corresponding fractal set A. It is more difficult to find conditions for the fk which ensure that A has a nice structure. We describe a technique which allows us to determine self-similar sets with a particularly simple structure. Some of the resulting examples are known, like the Sierpiński gasket and carpet, some others seem to be new. A simple structure is necessary when we want to do classical analysis on A; for instance, define harmonic functions and a Laplace operator. So far, much analysis has been realized on a very small class of fractal spaces–essentially the relatives of the Sierpiński gasket. In this chapter, we discuss two classes of infinitely ramified fractals which seem to be more realistic from the point of view of physical modeling, and we give examples for which fractal analysis seems to be possible. A property of the boundaries for these fractal classes is verified.