Abstract
We study the problem of the existence of increasing and continuous solutions \(\varphi :[0,1]\rightarrow [0,1]\) such that \(\varphi (0)=0\) and \(\varphi (1)=1\) of the functional equation
where \(N\in {\mathbb {N}}\) and \(f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]\) are strictly increasing contractions satisfying the following condition \(0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1\). In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.
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References
Billingsley, P.: Probability and Measure, Wiley Series in Probability and Mathematical Statistics, 3rd edn. Wiley, New York (1995)
Dubins, L.E., Freedman, D.A.: Invariant probabilities for certain Markov processes. Ann. Math. Stat. 37, 837–848 (1966)
Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Kania, T., Máthé, A., Morawiec, J., Rmoutil, M., Zürcher, T.: A functional equation (manuscript)
Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Applied Mathematical Sciences, vol. 97, 2nd edn. Springer, New York (1994)
Lasota, A., Myjak, J.: Generic properties of fractal measures. Bull. Pol. Acad. Sci. Math. 42, 283–296 (1994)
Lasota, A., Pianigiani, G.: Invariant measures on topological spaces. Boll. Un. Mat. Ital. B (5) 14, 592–603 (1977)
Matkowski, J.: Remark on BV-solutions of a functional equation connected with invariant measures. Aequationes Math. 29, 210–213 (1985)
Misiewicz, J., Wesołowski, J.: Winding planar probabilities. Metrika 75, 507–519 (2012)
Ngai, S.M., Wang, Y.: Self-similar measures associated to IFS with non-uniform contraction ratios. Asian J. Math. 9, 227–244 (2005)
Peres, Y., Schlag, W., Solomyak, B.: Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), pp. 39–65. Birkhäuser, Basel, Prog. Probab. 46 (2000)
Rényi, A.: Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)
Szarek, T.: Generic properties of learning systems. Ann. Polon. Math. 73, 93–103 (2000)
Szarek, T.: Invariant measures for iterated function systems. Ann. Polon. Math. 75, 87–98 (2000)
Acknowledgements
The research of the first author was supported by the Silesian University Mathematics Department (Iterative Functional Equations and Real Analysis program). Furthermore, the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 291497 while the second author was a postdoctoral researcher at the University of Warwick.
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Morawiec, J., Zürcher, T. On a problem of Janusz Matkowski and Jacek Wesołowski. Aequat. Math. 92, 601–615 (2018). https://doi.org/10.1007/s00010-018-0556-5
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DOI: https://doi.org/10.1007/s00010-018-0556-5