Bull. Austral. Math. Soc. 72(3) pp.441--454, 2005.

The Hutchinson-Barnsley theory
for infinite iterated function systems

Gertruda Gwóźdź-Łukawska

Jacek Jachymski

Received: 19th July, 2005

We are grateful to Andrzej Komisarski for some useful discussion.

Abstract

We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {F_i : i $\in$ $\mathbb {N}$ } is a family of Matkowski's contractions on a complete metric space (X, d ) such that (F_i x₀)_i is bounded for some x₀ $\in$ X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, K = $\bigcup \limits _{{i\in {\mathbb N}}}^{}$ F_i (K). Moreover, given $\si$ $\in$ $\mathbb {N}$ and x $\in$ X, the limit $\lim \limits _{{n\ra \iy }}^{}$ Fo ^...oF(x) exists and does not depend on x. We also study separately the case in which (X, d ) is Menger convex or compact. Finally, we answer a question posed by Máté concerning a finite iterated function system {F₁,..., F_N } with the property that each of F_i has a contractive fixed point.

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MathSciNet: MR2199645

Z'blatt-MATH: 1098.39015

References

J. Andres and J. Fišer;
Metric and topological multivalued fractals,
Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), pp. 1277--1289. MR2063892
J. Andres, J. Fišer, G. Gabor and K. Leśniak;
Multivalued fractals,
Chaos Solitons Fractals 24 (2005), pp. 665--700. MR2116280
J. Andres and L. Górniewicz;
On the Banach contraction principle for multivalued mappings,
in Approximation, optimization and mathematical economics (Pointe-à Pitre) (Physica, Heidelberg, 2001), pp. 1--23. MR1842872
M.F. Barnsley;
Fractals everywhere (Academic Press, New York, 1988). MR1231795
L.M. Blumenthal;
Theory and applications of distance geometry (Clarendon Press, Oxford, 1953). MR54981
F.E. Browder;
On the convergence of successive approximations for nonlinear functional equation,
Indag. Math. 30 (1968), pp. 27--35. MR230180
M. Edelstein;
On fixed and periodic points under contractive mappings,
J. London Math. Soc. 37 (1962), pp. 74--79. MR133102
R. Engelking;
General topology (Polish Scientific Publishers, Warszawa, 1977). MR500780
A. Granas and J. Dugundji;
Fixed point theory,
Springer Monographs in Mathematics (Springer-Verlag, New York, 2003). MR1987179
E. Hille and R.S. Phillips;
Functional analysis and semi-groups,
Amer. Math. Soc. Colloq. Publ. 31 (American Mathematical Society, Providence, R.I., 1957). MR89373
J. E. Hutchinson;
Fractals and self-similarity,
Indiana Univ. Math. J. 30 (1981), pp. 713--747. MR625600
J. Jachymski;
Equivalence of some contractivity properties over metrical structures,
Proc. Amer. Math. Soc. 125 (1997), pp. 2327--2335. MR1389524
J. Jachymski;
An extension of A. Ostrovski's theorem on the round-off stability of iterations,
Aequationes Math. 53 (1997), pp. 242--253. MR1444177
J. Jachymski, L. Gajek and P. Pokarowski;
The Tarski-Kantorovitch principle and the theory of iterated function systems,
Bull. Austral. Math. Soc. 61 (2000), pp. 247--261. MR1748704
L. Máté;
The Hutchinson-Barnsley theory for certain non-contraction mappings,
Period. Math. Hungar. 27 (1993), pp. 21--33. MR1258979
L. Máté;
On infinite composition of affine mappings,
Fund. Math. 159 (1999), pp. 85--90. MR1669710
J. Matkowski;
Integrable solutions of functional equations,
(Dissertationes Math.) 127 (Rozprawy Mat., Warszawa, 1975). MR412650
J. Matkowski;
Fixed point theorem for mappings with a contractive iterate at a point,
Proc. Amer. Math. Soc. 62 (1977), pp. 344--348. MR436113
J. Matkowski;
Nonlinear contractions in metrically convex space,
Publ. Math. Debrecen 45 (1993), pp. 103--114. MR1291805

ISSN 0004-9727

Bulletin of the Australian Mathematical Society

The Hutchinson-Barnsley theory for infinite iterated function systems