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 = sup i 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.
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References
Bandt C., Self-similar sets. I. Topological Markov chains and mixed self-similar sets, Math. Nachr., 1989, 142, 107–123
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
Falconer K.J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986
Falconer K., Fractal Geometry, John Wiley & Sons, Chichester, 1990
Hille M.R., Remarks on limit sets of infinite iterated function systems, Monatsh. Math., 2012, 168(2), 215–237
Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747
Kravchenko A.S., Completeness of the space of separable measures in the Kantorovich-Rubinshtein metric, Siberian Math. J., 2006, 47(1), 68–76
Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math., 44, Cambridge University Press, Cambridge, 1995
Mauldin R.D., Infinite iterated function systems: theory and applications, In: Fractal Geometry and Stochastics, Progr. Probab., 37, Birkhäuser, Basel, 1995, 91–110
Mauldin R.D., Urbanski M., Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 1996, 73(1), 105–154
Mauldin R.D., Williams S.C., Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc., 1986, 295(1), 325–346
Mihail A., Miculescu R., The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 2009, 11(61)(1), 21–32
Secelean N.A., The existence of the attractor of countable iterated function systems, Mediterr. J. Math., 2012, 9(1), 61–79
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Barrozo, M.F., Molter, U. Countable contraction mappings in metric spaces: invariant sets and measure. centr.eur.j.math. 12, 593–602 (2014). https://doi.org/10.2478/s11533-013-0371-0
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DOI: https://doi.org/10.2478/s11533-013-0371-0