## Self-similar measures for iterated function systems driven by weak contractions.(English)Zbl 1396.28018

Summary: We show the existence and uniqueness for self-similar measures for iterated function systems driven by weak contractions. Our main idea is using the duality theorem of Kantorovich-Rubinstein and equivalent conditions for weak contractions established by Jachymski. We also show collage theorems for such iterated function systems.

### MSC:

 28A80 Fractals 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

 [1] J. Andres and J. Fišer, Metric and topological multivalued fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 4, 1277-1289. · Zbl 1057.28003 [2] M. F. Barnsley, Existence and uniqueness of orbital measures.v1. [3] M. F. Barnsley, Superfractals, Cambridge University Press, Cambridge, 2006. [4] M. F. Barnsley, V. Ervin, D. Hardin and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 7, 1975-1977. · Zbl 0613.28008 [5] F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 27-35. [6] M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), no. 4, 99-102. · Zbl 0573.54033 [7] M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), no. 2, 381-414. · Zbl 0608.28003 [8] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. · Zbl 0598.28011 [9] J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2327-2335. · Zbl 0887.47039 [10] L. V. Kantorovič and G. Š. Rubinšteĭn, On a space of completely additive functions, Vestnik Leningrad. Univ. 13 (1958), no. 7, 52-59. [11] M. Kesseböhmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski’s question mark function, J. Number Theory 128 (2008), no. 9, 2663-2686. · Zbl 1154.28001 [12] M. A. Krasnosel’skiĭ and V. Ja. Stecenko, On the theory of concave operator equations, Sibirsk. Mat. Ž. 10 (1969), 565-572. [13] K. Leśniak, Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 1, 1-8. · Zbl 1108.47048 [14] H. Minkowski, Zur Geometrie der Zahlen, in Verhandlungen des dritten Internationalen Mathematiker-Kongresses (Heidelberg, 1904), 164-173, Druck und Verlag von B. G. Teubner, Leipzig, 1905; available at https://faculty.math.illinois.edu/ reznick/Minkowski.pdf. [15] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683-2693. · Zbl 1042.47521 [16] C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. [17] K. R. Wicks, Fractals and hyperspaces, Lecture Notes in Mathematics, 1492, Springer-Verlag, Berlin, 1991. · Zbl 0770.54048
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