2007/03/14 by Michael F. Barnsley, Barnsley, Michael F. · 1 citation
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #advanced mathematical theories #math.DS
paper · pdf · doi:10.48550/arxiv.math/0703398
18 Pages, 7 figures
arxiv created 2007/03/14 · openalex publication_date 2007/03/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is in the form of an essay. It defines fractal tops and code space structures associated with set-attractors of hyperbolic iterated function 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 A of the first IFS and whose range is contained in the attractor B of the second IFS. This transformation 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 between A and B when the code space structures are the same. Conversely, if two IFS are homeomorphic then they possess the same code space structure. 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.