Abstract
Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of mapsw j :K →K,j=1, 2,⋯,N, whereK is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsleyet al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. F. Barnsley (1986):Fractal functions and interpolation. Constr. Approx.,2:303–329.
M. F. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A,399:243–275.
M. F. Barnsley, J. Elton (1988):A new class of Markov processes for image encoding. Adv. in Appl. Probab.,20:14–32.
M. F. Barnsley, V. Ervin, D. P. Hardin, J. Lancaster (1986):Solution of an inverse problem for fractals and other sets. Proc. Nat. Acad. Sci. U.S.A.,83:1975–1977.
M. F.Barnsley, J.Elton, D.Hardin, P.Massopust (preprint): Hidden Variable Fractal Interpolation Functions.
M. F. Barnsley, J. S. Geronimo, A. N. Harrington (1984):Geometry and combinatorics of Julia sets of real quadratic maps. J. Statist. Phys.,37:51–92.
M. F.Barnsley, A. D.Sloan (1986) (preprint):Image compression.
T.Bedford Dimension and dynamics for fractal recurrent sets.
M.Berger, Y.Amit (preprint):Products of random affine maps.
F. M.Dekking (1982): Recurrent Sets: A Fractal Formalism. Delft University of Technology.
J. Dugundji (1966): Topology. Boston: Allyn and Bacon, p. 253.
J. Elton (1987):An ergodic theorem for iterated maps. Ergodic Theory Dynamical Systems,7:481–488.
W. Feller (1957): An Introduction to Probability Theory and Its Applications. London: Wiley.
J. Hutchinson (1981):Fractals and self-similarity. Indiana Univ. Math. J.,30:731–747.
D. P. Hardin, P. Massopust (1986):Dynamical systems arising from iterated function systems. Comm. Math. Phys.,105:455–460.
U. Krengel (1985): Ergodic Theorems, New York: de Gruyter.
T. Li, J. A. Yorke (1975):Period three implies chaos. Amer. Math. Monthly,82:985–992.
E. Seneta (1973): Non-negative Matrices. New York: Wiley.
Author information
Authors and Affiliations
Additional information
Communicated by Edward B. Saff.
Rights and permissions
About this article
Cite this article
Barnsley, M.F., Elton, J.H. & Hardin, D.P. Recurrent iterated function systems. Constr. Approx 5, 3–31 (1989). https://doi.org/10.1007/BF01889596
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01889596