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Continuity of Attractors and Invariant Measures for Iterated Function Systems

Published online by Cambridge University Press:  20 November 2018

P. M. Centore  and
E. R. Vrscay
E. R. Vrscay
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G1
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Abstract

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We prove the "folklore" results that both the attractor A and invariant measure μ of an N-map Iterated Function System (IFS) vary continuously with variations in the contractive IFS maps as well as the probabilities. This represents a generalization of Barnsley's result showing the continuity of attractors with respect to variations of a parameter appearing in the IFS maps. Some applications are presented, including approximations of attractors and invariant measures of nonlinear IFS, as well as some novel approximations of Julia sets for quadratic complex maps.

Keywords

Primary: 28A33secondary: 30D0558F11
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994 , pp. 315 - 329
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Barnsley, M. F., Fractals Everywhere, Academic Press, New York, 1988.Google Scholar
2. Hutchinson, J., Fractals and self-similarity, Indiana Univ. Math. J. 30(1981), 713–747.Google Scholar
3. Barnsley, M. F. and Demko, S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399(1985), 243–275.Google Scholar
4. Vrscay, E. R. and Weil, D., “Missing moment”, and perturbative methods for polynomial iterated function systems, Phys. D 50(1991), 478–492.Google Scholar
5. Barnsley, M. F., Demko, S. G., J. Elton and Geronimo, J. S., Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. H. Poincaré 24(1988), 367– 394.Google Scholar
6. Vrscay, E. R., Iterated Functions Systems: theory, applications and the inverse problem, Proceedings of the NATO Advanced Study Institute on Fractal Geometry and Analysis, July 3-21, 1989, Montréal, Canada, (eds. J. Belair and S. Dubuc), (1991), 405-468.Google Scholar
7. Brolin, H., Invariant sets under iteration of rational functions, Ark. Mat. 6(1966), 103–144.Google Scholar
8. Blanchard, P., Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11(1984), 85– 141.Google Scholar
9. Devaney, R., An introduction to Chaotic Dynamical Systems, Benjamin-Cummings, Menlo Park, California, 1986.Google Scholar
10. Ulam, S. M. and von Neumann, J., On combinations of stochastic and deterministic processes, Bull. Amer. Math. Soc. 53(1947), 1120.Google Scholar
11. Cabrelli, C. A., Forte, B., Molter, U. M. and Vrscay, E. R., Iterated Fuzzy Set Systems: A new approach to the inverse problem for fractals and other sets, J. Math. Anal. Appl. 171(1992), 79–100.Google Scholar
12. Forte, B., Schiavo, M. L. and Vrscay, E. R., Continuity properties of attractors of Iterated Fuzzy Set Systems, J. Australian Mathematical Society, Servies B, to appear.Google Scholar
13. Forte, B. and Vrscay, E. R., Solving the inverse problem for function and image approximation using iterated function systems, in preparation. See also Solving the inverse problem for function/image approximation using iterated function systems I. theoretical basis. Fractals 2(1994) 325-334. Solving the inverse problem for function/image approximation using iterated function systems II. algorithm and computations. Fractals 2(1994) 335–346.Google Scholar