Transformations between Self-Referential Sets

1. INTRODUCTION. It is well known that there exist functions such as Peano curves which map a closed real interval continuously onto a filled square. But did you know that there are continuous transformations from a fractal fern onto the square? Moreover it is straightforward to make pictures of such transformations by means of a chaos game. We say that a compact subset of R 2 is self-referential if it can be written as a union of contractive transformations applied to itself. For example, a filled square can be written as the union of four smaller filled rectangles; we can think of these smaller rectangles as providing an overlapping tiling of the original square. Similarly, a filled triangle can be written, in many ways, as the union of four smaller filled triangles. Also a fractal fern, as illustrated in Figure 5, may be the union of four affinely transformed copies of itself. In each case, as we will explain, the overlapping tiling provides an addressing structure and a natural dynamical system on the tiled region. In this article we show that if two self-referential sets have the same addressing structure then there exists a natural homeomorphism between them; moreover, their dynamical systems are conjugate. Actually, we show how to construct transformations between diverse self-referential sets. We call these transformations fractal because they may be non-differentiable and they may change the Hausdorff dimensions of sets upon which they act. Fractal transformations are very beautiful and, I believe, worthy of the attention of the mathematics community. They exemplify basic notions in topology, probability, dynamical systems, and geometry. They may be applied to computer graphics to produce digital content with new look-and-feel; they may also be relevant to image compression and biological modelling [2]. But their most important credential is that they are mathematically intriguing. 2. HYPERBOLIC ITERATED FUNCTION SYSTEMS. In this section we explain what is meant by an IFS, its attractor, and the associated code space. We observe that there is a continuous mapping from the code space onto the attractor. Then we introduce a special example.