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THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM

Part of: Classical measure theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  11 June 2013

MICHAEL F. BARNSLEY  and
ANDREW VINCE
MICHAEL F. BARNSLEY
Affiliation:
Department of Mathematics, Australian National University, Canberra, ACT, Australia email michael.barnsley@maths.anu.edu.aumbarnsley@aol.com
ANDREW VINCE*
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA
*
✉avince@ufl.edu
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Abstract

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We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas of Kieninger [Iterated Function Systems on Compact Hausdorff Spaces (Shaker, Aachen, 2002)] and McGehee and Wiandt [‘Conley decomposition for closed relations’, Differ. Equ. Appl. 12 (2006), 1–47] restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas.

Keywords

iterated function systemattractor–repeller

MSC classification

Secondary: 37C70: Attractors and repellers, topological structure 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 267 - 279
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Atkins, R., Barnsley, M. F., Wilson, D. C. and Vince, A., ‘A characterization of point-fibred affine iterated function systems’, Topology Proc. 38 (2010), 189–211.Google Scholar
Barnsley, M. F., Fractals Everywhere (Academic Press, Boston, MA, 1988).Google Scholar
Barnsley, M. F., ‘Fractal image compression’, Notices Amer. Math. Soc. 43 (1996), 657–662.Google Scholar
Barnsley, M. F. and Lesniak, K., ‘On the continuity of the Hutchinson operator’, arXiv:1202.2485.Google Scholar
Barnsley, M. F. and Vince, A., ‘The chaos game on a general iterated function system’, Ergod. Th. & Dynam. Sys. 31 (2011), 1073–1079.CrossRefGoogle Scholar
Barnsley, M. F. and Vince, A., ‘Real projective iterated function systems’, J. Geom. Anal. 22 (2012), 1137–1172.CrossRefGoogle Scholar
Conley, C., ‘Isolated invariant sets and the Morse index’, CBMS Regional Conference Series, 38 (American Mathematical Society, 1978).Google Scholar
Edgar, G. A., Measure, Topology, and Fractal Geometry (Springer, New York, 1990).CrossRefGoogle Scholar
Hirsch, M. W. and Hurley, M., ‘Connected components of attractors and other stable sets’, Aequationes Math. 53 (1997), 308–323.CrossRefGoogle Scholar
Hutchinson, J., ‘Fractals and self-similarity’, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
Kameyama, A., Distances on Topological Self-similar Sets, Proceedings of Symposia in Pure Mathematics, 72.1 (2004).CrossRefGoogle Scholar
Kieninger, B., Iterated Function Systems on Compact Hausdorff Spaces (Shaker, Aachen, 2002).Google Scholar
McGehee, R., ‘Attractors for closed relations on compact Hausdorff spaces’, Indiana Univ. Math. J. 41 (1992), 1165–1209.CrossRefGoogle Scholar
McGehee, R. P. and Wiandt, T., ‘Conley decomposition for closed relations’, Differ. Equ. Appl. 12 (2006), 1–47.CrossRefGoogle Scholar
Milnor, J., ‘On the concept of attractor’, Commun. Math. Phys. 99 (1985), 177–195.CrossRefGoogle Scholar
Milnor, J., ‘On the concept of attractor: correction and remarks’, Commun. Math. Phys. 102 (1985), 517–519.CrossRefGoogle Scholar
Vince, A., ‘Möbius iterated function systems’, Trans. Amer. Math. Soc. 265 (2013), 491–509.Google Scholar