Transformations between attractors of hyperbolic iterated function systems

This paper is in the form of an essay. It defines fractal tops and code space structures associated with set-attractors of hyperbolic iterated func- tion systems (IFSs). The fractal top of an IFS is associated with a certain shift invariant subspace of code space, whence the entropy of the IFS, and of its set-attractor, may be defined. Given any ordered pair of hyperbolic IFSs, each with the same number of maps, there is a natural transformation, constructed with the aid of fractal tops, whose domain is the attractor A1of the first IFS and whose range is contained in the attractor A2 of the second IFS. This trans- formation is continuous when the code space structure of the IFS is "contained in" the code space structure of the second IFS, and is a homeomorphism be- tween A1 and A2 when the code space structures are the same. Conversely, if two IFS are homeomorphic then they possess the same code space struc- ture. Hence we obtain that two IFS attractors are homeomorphic then they have the same entropy. Several examples of fractal transformations and fractal homeomphisms are given.