Countable contraction mappings in metric spaces: invariant sets and measure

• Autorzy: María Barrozo , Ursula Molter
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Contraction maps; Countable iterated function system; Invariant set; Invariant measure

Kategorie tematyczne

• 37C70: Attractors and repellers, topological structure
• 37C25: Fixed points, periodic points, fixed-point index theory
• 28A80: Fractals

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 4
• Strony: 593-602

Daty

wydano

2014-04-01

online

2014-01-17

Twórcy

autor

María Barrozo

autor

Ursula Molter

Bibliografia

• [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
• [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
• [3] Falconer K.J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986
• [4] Falconer K., Fractal Geometry, John Wiley & Sons, Chichester, 1990
• [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
• [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
• [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] 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] 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
• [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
• [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
• [12] Mihail A., Miculescu R., The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 2009, 11(61)(1), 21–32
• [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

Identyfikatory

DOI

10.2478/s11533-013-0371-0