# Countable contraction mappings in metric spaces: invariant sets and measure

Open Mathematics (2014)

• Volume: 12, Issue: 4, page 593-602
• ISSN: 2391-5455

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## Abstract

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We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.

## How to cite

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María Barrozo, and Ursula Molter. "Countable contraction mappings in metric spaces: invariant sets and measure." Open Mathematics 12.4 (2014): 593-602. <http://eudml.org/doc/269455>.

@article{MaríaBarrozo2014,
abstract = {We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = \{F i: i ∈ ℕ\}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = \{ρ k\}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.},
author = {María Barrozo, Ursula Molter},
journal = {Open Mathematics},
keywords = {Contraction maps; Countable iterated function system; Invariant set; Invariant measure; contraction maps; countable iterated function system; invariant set; invariant measure},
language = {eng},
number = {4},
pages = {593-602},
title = {Countable contraction mappings in metric spaces: invariant sets and measure},
url = {http://eudml.org/doc/269455},
volume = {12},
year = {2014},
}

TY - JOUR
AU - María Barrozo
AU - Ursula Molter
TI - Countable contraction mappings in metric spaces: invariant sets and measure
JO - Open Mathematics
PY - 2014
VL - 12
IS - 4
SP - 593
EP - 602
AB - We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.
LA - eng
KW - Contraction maps; Countable iterated function system; Invariant set; Invariant measure; contraction maps; countable iterated function system; invariant set; invariant measure
UR - http://eudml.org/doc/269455
ER -

## References

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1. [1] Bandt C., Self-similar sets. I. Topological Markov chains and mixed self-similar sets, Math. Nachr., 1989, 142, 107–123 http://dx.doi.org/10.1002/mana.19891420107 Zbl0707.28004
2. [2] Barnsley M.F., Demko S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 1985, 399(1817), 243–275 http://dx.doi.org/10.1098/rspa.1985.0057 Zbl0588.28002
3. [3] Falconer K.J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986 Zbl0587.28004
4. [4] Falconer K., Fractal Geometry, John Wiley & Sons, Chichester, 1990
5. [5] Hille M.R., Remarks on limit sets of infinite iterated function systems, Monatsh. Math., 2012, 168(2), 215–237 http://dx.doi.org/10.1007/s00605-011-0357-6 Zbl1272.28006
6. [6] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747 http://dx.doi.org/10.1512/iumj.1981.30.30055 Zbl0598.28011
7. [7] Kravchenko A.S., Completeness of the space of separable measures in the Kantorovich-Rubinshtein metric, Siberian Math. J., 2006, 47(1), 68–76 http://dx.doi.org/10.1007/s11202-006-0009-6
8. [8] Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math., 44, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511623813
9. [9] Mauldin R.D., Infinite iterated function systems: theory and applications, In: Fractal Geometry and Stochastics, Progr. Probab., 37, Birkhäuser, Basel, 1995, 91–110 http://dx.doi.org/10.1007/978-3-0348-7755-8_5 Zbl0841.28009
10. [10] Mauldin R.D., Urbanski M., Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 1996, 73(1), 105–154 http://dx.doi.org/10.1112/plms/s3-73.1.105 Zbl0852.28005
11. [11] Mauldin R.D., Williams S.C., Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc., 1986, 295(1), 325–346 http://dx.doi.org/10.1090/S0002-9947-1986-0831202-5 Zbl0625.54047
12. [12] Mihail A., Miculescu R., The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 2009, 11(61)(1), 21–32 Zbl1199.28030
13. [13] Secelean N.A., The existence of the attractor of countable iterated function systems, Mediterr. J. Math., 2012, 9(1), 61–79 http://dx.doi.org/10.1007/s00009-011-0116-x Zbl1242.28014

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