Iterated function system

A family of mappings, where is a complete metric space. Usually, is a finite set, . An iterated function system is called hyperbolic if all are contracting (cf. Contraction) for . An iterated function system induces a mapping from the space to itself by . If the iterated function system is hyperbolic and one restricts to the space of non-empty closed bounded sets equipped with the Hausdorff metric, then it follows from the contracting-mapping principle that has a unique fixed point ; moreover, is compact ([a8], [a7]). The set is called the attractor, or invariant set, of the iterated function system. If is a Euclidean space and the are similitudes, then the attractor is called a self-similar set. These sets are usually fractals (cf. Fractals); an example is the triadic Cantor set (cf. also Cantor set), which is the attractor of the iterated function system

A generalization is the concept of a recurrent self-similar (or mixed self-similar) set generated by a recurrent iterated function system. Let be an -matrix of zeros and ones satisfying for some , and let contracting mappings be given for each such that . Then (see [a3] or [a1]) there is a unique vector of non-empty compact sets such that for ,

An iterated function system with probabilities is an iterated function system together with a probability vector (i.e., and ). This induces a mapping on the space of Borel probability measures by for all Borel sets (cf. also Borel set) . If the iterated function system is hyperbolic, then there is a unique fixed point for , i.e. an invariant measure, whose topological support is the attractor of the iterated function system [a7]. Convergence to is also obtained by associating a random dynamical system with the iterated function system via the Markov chain , where the form a sequence of independent random variables with distributions [a2]. The same type of result holds for recurrent iterated function systems, taking for a Markov chain on whose transition probabilities are positive if and only if the are positive [a3], [a4]. Iterated function systems are applied for approximation purposes (see, e.g., [a5]) and the construction of wavelets (see, e.g., [a6]).

References

[a1]	C. Bandt, "Self similar sets I. Topological Markov chains and mixed self-similar sets" Math. Nachr. , 142 (1989) pp. 107–123
[a2]	M.F. Barnsley, D. Demko, "Iterated function systems and the global construction of fractals" Proc. Roy. Soc. London A , 399 (1985) pp. 243–275
[a3]	M.F. Barnsley, J.H. Elton, D.P. Hardin, "Recurrent iterated function systems" Constr. Approx. , 5 (1989) pp. 3–31
[a4]	J. Elton, "An ergodic theorem for iterated maps" Ergodic Th. & Dynamical Systems , 7 (1987) pp. 481–488
[a5]	B. Forte, E.R. Vrscay, "Solving the inverse problem for measures using iterated function systems: a new approach" Adv. Appl. Probab. , 27 (1995) pp. 800–820
[a6]	J.S. Geronimo, D.P. Hardin, P.R. Massopust, "Fractal functions and wavelet expansions based on several scaling functions" J. Approx. Th. , 78 (1994) pp. 373–401
[a7]	J.E. Hutchinson, "Fractals and self-similarity" Indiana Univ. Math. J. , 30 (1981) pp. 713–747
[a8]	R.F. Williams, "Composition of contractions" Bol. Soc. Brasil. Mat. , 2 (1971) pp. 55–59