Geodesic
Continuity of Attractors and Invariant Measures for Iterated Function Systems
Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 315-329

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1994-048-6
Mots-clés : Primary: 28A33secondary: 30D05, 58F11 
Vrscay, E. R. Continuity of Attractors and Invariant Measures for Iterated Function Systems. Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 315-329. doi: 10.4153/CMB-1994-048-6
@article{10_4153_CMB_1994_048_6,
     author = {Vrscay, E. R.},
     title = {Continuity of {Attractors} and {Invariant} {Measures} for {Iterated} {Function} {Systems}},
     journal = {Canadian mathematical bulletin},
     pages = {315--329},
     year = {1994},
     volume = {37},
     number = {3},
     doi = {10.4153/CMB-1994-048-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1994-048-6/}
}
TY  - JOUR
AU  - Vrscay, E. R.
TI  - Continuity of Attractors and Invariant Measures for Iterated Function Systems
JO  - Canadian mathematical bulletin
PY  - 1994
SP  - 315
EP  - 329
VL  - 37
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1994-048-6/
DO  - 10.4153/CMB-1994-048-6
ID  - 10_4153_CMB_1994_048_6
ER  - 
%0 Journal Article
%A Vrscay, E. R.
%T Continuity of Attractors and Invariant Measures for Iterated Function Systems
%J Canadian mathematical bulletin
%D 1994
%P 315-329
%V 37
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1994-048-6/
%R 10.4153/CMB-1994-048-6
%F 10_4153_CMB_1994_048_6

[1] 1. Barnsley, M. F., Fractals Everywhere, Academic Press, New York, 1988. Google Scholar

[2] 2. Hutchinson, J., Fractals and self-similarity, Indiana Univ. Math. J. 30(1981), 713–747. Google Scholar

[3] 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] 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] 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] 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] 7. Brolin, H., Invariant sets under iteration of rational functions, Ark. Mat. 6(1966), 103–144. Google Scholar

[8] 8. Blanchard, P., Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11(1984), 85– 141. Google Scholar

[9] 9. Devaney, R., An introduction to Chaotic Dynamical Systems, Benjamin-Cummings, Menlo Park, California, 1986. Google Scholar

[10] 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] 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] 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] 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

Cité par Sources :