Some Recent Progress Concerning Topology of Fractals

Michael F. Barnsley (), David C. Wilson () and Krzysztof Leśniak ()
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Michael F. Barnsley: Australian National University, Department of Mathematics
David C. Wilson: University of Florida, Department of Mathematics
Krzysztof Leśniak: Nicolaus Copernicus University, Faculty of Mathematics and Computer Science

A chapter in Recent Progress in General Topology III, 2014, pp 69-92 from Springer

Abstract: Abstract We give a panorama of recent results in fractal geometry, concerning properties of attractors of iterated function systems (IFS). The work we present has a topological flavour but is motivated mainly by geometry. We review the various notions of attractor, then introduce the symbolic dynamics language for handling IFS and review its application to the exciting new area of fractal homeomorphisms. We also recall Kameyama’s question on topological contractivity of reasonable IFSs, and discuss IFS theory from the point of view of Conley decompositions. We review the recently discovered theory of projective IFSs, which live in beautiful environments, projective spaces, with their own specific structures that generalize affine IFS theory. We conclude by explaining the random iteration algorithm from a topological point of view. This algorithm, which often reveals attractors, works due to an intimate connection between the tree structure of code space and the stochastic process that generates the code. The result is a deterministic version of the "chaos game" which always works and avoids probabilistic notions.

Keywords: Iterate Function System; Symbolic Dynamic; Compact Hausdorff Space; Baire Category; Code Space (search for similar items in EconPapers)
Date: 2014
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DOI: 10.2991/978-94-6239-024-9_2

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