Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 084, 21 pages      arXiv:1308.3819

Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems

Michael F. Barnsley a and Andrew Vince b
a) Mathematical Sciences Institute, Australian National University, Australia
b) Department of Mathematics, Univesity of Florida, USA

Received June 23, 2015, in final form October 13, 2015; Published online October 16, 2015

The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.

Key words: iterated function system; fast basins; fractal continuation; fractal manifold.

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  1. Barnsley M.F., Transformations between self-referential sets, Amer. Math. Monthly 116 (2009), 291-304, math.DS/0703398.
  2. Barnsley M.F., Leśniak K., Basic topological structure of fast basins, arXiv:1308.4230.
  3. Barnsley M.F., Leśniak K., Vince A., Symbolic iterated function systems, fast basins and fractal manifolds, arXiv:1308.3819v1.
  4. Barnsley M.F., Vince A., The chaos game on a general iterated function system, Ergodic Theory Dynam. Systems 31 (2011), 1073-1079, arXiv:1005.0322.
  5. Barnsley M.F., Vince A., Developments in fractal geometry, Bull. Math. Sci. 3 (2013), 299-348.
  6. Barnsley M.F., Vince A., Fractal continuation, Constr. Approx. 38 (2013), 311-337, arXiv:1209.6100.
  7. Barnsley M.F., Vince A., The Conley attractors of an iterated function system, Bull. Aust. Math. Soc. 88 (2013), 267-279, arXiv:1206.6319.
  8. Barnsley M.F., Vince A., Fractal tilings from iterated function systems, Discrete Comput. Geom. 51 (2014), 729-752, arXiv:1310.6344.
  9. Hata M., On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381-414.
  10. Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
  11. Ionescu M., Kumjian A., Groupoid actions on fractafolds, SIGMA 10 (2014), 068, 14 pages, arXiv:1311.3880.
  12. Kieninger B., Iterated function systems on compact Hausdorff spaces, Ph.D. Thesis, Augsburg University, Germany, 2002.
  13. Mandelbrot B.B., The fractal geometry of nature, W.H. Freeman and Co., San Francisco, Calif., 1982.
  14. Strichartz R.S., Fractals in the large, Canad. J. Math. 50 (1998), 638-657.
  15. Strichartz R.S., Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 4019-4043.
  16. Strichartz R.S., Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006.
  17. Vince A., Möbius iterated function systems, Trans. Amer. Math. Soc. 365 (2013), 491-509.

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