Abstract
Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Mobius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor–repeller pairs and the Conley “landscape picture” for an IFS.
Highlights
Metric spaces such as Euclidean space, the sphere, and projective space possess rich families of simple geometrical transformations f : X → X
Many textbooks use pictures of such objects to illustrate the idea of a fractal; Fig. 1 illustrates a few familiar iterated function systems (IFSs) fractals, and some newer fractal objects associated with simple geometrical IFSs
In so doing we introduce the concept of an attractor–repeller pair that is explained in more detail in Sect. 12, which deals with Conley’s “landscape picture” for an IFS
Summary
Metric spaces such as Euclidean space, the sphere, and projective space possess rich families of simple geometrical transformations f : X → X. In this example, we change the space X to be R2 ∪ {∞}, the one point compactification of R2, where ∞ is “the point at infinity”, and we define fn(∞) = ∞, the basin is no longer the whole space and the dual repeller is A∗ = {∞} Remarks analogous to those mentioned in this paragraph hold, for all similitude IFSs. Note that S := R2 ∪ {∞} can be represented as a sphere (with infinity at say the north pole) and R2 is embedded in S by stereographic projection. The IFS (2.2) is invertible and its inverse F−1 possesses a unique attractor, equal to the dual repeller of F, the spiral Cantor set to which we have just referred. For further details on such Möbius examples see [124]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
