On a problem of Janusz Matkowski and Jacek Wesołowski, II

  • Janusz MorawiecEmail author
  • Thomas Zürcher
Open Access


We continue our study started in Morawiec and Zürcher (Aequ Math 92(4):601–615, 2018) of the functional equation
$$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _{n=0}^{N}\varphi (f_n(0)) \end{aligned}$$
and its increasing and continuous solutions \(\varphi :[0,1]\rightarrow [0,1]\) such that \(\varphi (0)=0\) and \(\varphi (1)=1\). In this paper we assume that \(f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]\) are strictly increasing contractions such that
$$\begin{aligned} 0\le f_0(0)<f_0(1)\le f_1(0)<\cdots<f_{N-1}(1)\le f_N(0)<f_N(1)\le 1 \end{aligned}$$
and at least one of the weak inequalities is strong.


Functional equations Iterated function systems Singular functions Absolutely continuous functions 

Mathematics Subject Classification

Primary 39B12 Secondary 37A05 



  1. 1.
    Barnsley, M.: Fractals Everywhere. Academic Press Inc., Boston (1988)zbMATHGoogle Scholar
  2. 2.
    Biswas, H.R.: Ergodic theory and mixing properties. J. Pure Appl. Math. Adv. Appl. 12(1), 1–24 (2014)MathSciNetGoogle Scholar
  3. 3.
    Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011)zbMATHGoogle Scholar
  4. 4.
    Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997)zbMATHGoogle Scholar
  5. 5.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kannan, R., Krueger, C.K.: Advanced analysis on the real line. Universitext, Springer, New York (1996)Google Scholar
  7. 7.
    Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics. Volume 97 of Applied Mathematical Sciences, 2nd edn. Springer, New York (1994)zbMATHGoogle Scholar
  8. 8.
    Lasota, A., Myjak, J.: Generic properties of fractal measures. Bull. Polish Acad. Sci. Math. 42(4), 283–296 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Matkowski, J.: Remark on BV-solutions of a functional equation connected with invariant measures. Aequ. Math. 29(2–3), 210–213 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Morawiec, J., Zürcher, T.: Attractor of cantor type with positive measure. Results Math. 73(2), 1–13 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Morawiec, J., Zürcher, T.: On a problem of Janusz Matkowski and Jacek Wesołowski. Aequ. Math. 92(4), 601–615 (2018)CrossRefzbMATHGoogle Scholar
  12. 12.
    Phelps, R.R.: Lectures on Choquet’s Theorem. Volume 1757 of Lecture Notes in Mathematics, 2nd edn. Springer, Berlin (2001)Google Scholar
  13. 13.
    Wise, G.L., Hall, E.B.: Counterexamples in Probability and Real Analysis. The Clarendon Press, Oxford University Press, New York (1993)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Instytut MatematykiUniwersytet ŚląskiKatowicePoland

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