Proceedings of the Japan Academy, Series A, Mathematical Sciences

Self-similar measures for iterated function systems driven by weak contractions

Kazuki Okamura

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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.

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Proc. Japan Acad. Ser. A Math. Sci. Volume 94, Number 4 (2018), 31-35.

First available in Project Euclid: 5 April 2018

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Primary: 28A80: Fractals [See also 37Fxx] 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Self-similar measures iterated function systems weak contractions Kantorovich-Rubinstein duality theorem


Okamura, Kazuki. Self-similar measures for iterated function systems driven by weak contractions. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 4, 31--35. doi:10.3792/pjaa.94.31.

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  • J. Andres and J. Fišer, Metric and topological multivalued fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 4, 1277–1289.
  • M. F. Barnsley, Existence and uniqueness of orbital measures.v1.
  • M. F. Barnsley, Superfractals, Cambridge University Press, Cambridge, 2006.
  • 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.
  • 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.
  • M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), no. 4, 99–102.
  • M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), no. 2, 381–414.
  • J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.
  • J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2327–2335.
  • L. V. Kantorovič and G. Š. Rubinšteĭn, On a space of completely additive functions, Vestnik Leningrad. Univ. 13 (1958), no. 7, 52–59.
  • 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.
  • M. A. Krasnosel'skiĭ and V. Ja. Stecenko, On the theory of concave operator equations, Sibirsk. Mat. Ž. 10 (1969), 565–572.
  • K. Leśniak, Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 1, 1–8.
  • 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
  • B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683–2693.
  • C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.
  • K. R. Wicks, Fractals and hyperspaces, Lecture Notes in Mathematics, 1492, Springer-Verlag, Berlin, 1991.