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
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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
,
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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 |
Iterated function system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iterated_function_system&oldid=15812