A computer scientist's perspective on approximation of IFS invariant sets and measures with the random iteration algorithm

• Autorzy: Martyn Tomasz

Identyfikatory

DOI

10.24425/ijet.2024.152514

• Języki publikacji: EN

Abstrakty

EN

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

Słowa kluczowe

EN

IFS; discrete space; Markov chain; approximation; invariant set; invariant measure

• Wydawca: Polish Academy of Sciences, Committee of Electronics and Telecommunication
• Czasopismo: International Journal of Electronics and Telecommunications
• Rocznik: 2024
• Tom: Vol. 70, No. 4
• Strony: 1113--1123
• Opis fizyczny: Bibliogr., 14 poz.

Twórcy

autor

Martyn Tomasz

• Warsaw University of Technology

Bibliografia

• [1] M.F. Barnsley, Fractals everywhere, 2nd ed., Boston Academic Press, 1993.
• [2] M. Peruggia, Discrete Iterated Function Systems. A. K. Peters Wellesley MA, 1993.
• [3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numer-ical Recipes in C (2nd Ed.), Cambridge University Press, 1992.
• [4] J. Harrison, ”A machine-checked theory of floating point arithmetic”. In: Bertot Y, Dowek G, Th´ery L, Hirschowitz A, Paulin C (eds). Theorem Proving in Higher Order Logics. TPHOLs 1999. Lecture Notes in Computer Science, vol. 1690. Springer, Berlin, Heidelberg, 1999, pp. 113-130.
• [5] J-M. Muller, ”On the definition of ulp(x)”, Research Report, INRIA, LIP, 2005.
• [6] P. Diaconis and D. Freedman, ”Iterated Random Functions”, SIAM Rev., vol. 41, pp. 45-76, 1999.
• [7] Ö. Stenflo, ”Uniqueness of invariant measures for place-dependent random iterations of functions”, IMA Vol. Math. Appl., vol. 132, pp. 13-32, 2002.
• [8] M.F. Barnsley, S.G. Demko, J.H. Elton and J.S. Geronimo, ”Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities”, Ann. Inst. H.Poincar´e Probab. Statist., vol. 24, pp. 367-394, 1988.
• [9] J.H. Elton, ”An ergodic theorem for iterated maps”, Ergodic Theory and Dynam. Systems, vol. 7, pp. 481-488, 1987.
• [10] R.F. Serfozo, Basics of Applied Stochastic Processes, Springer-Verlag New York-Berlin, 2009.
• [11] D.W. Stroock DW, An Introduction to Markov Processes, Springer-Verlag New York-Berlin, 2005.
• [12] T. Martyn, ”The discrete charm of iterated function systems”. [Online]. Available: https://doi.org/10.48550/arXiv.2410.15139
• [13] W. Feller, An Introduction to Probability Theory and Its Applications (3th Ed.), Volume I, John Wiley & Sons, 1968.
• [14] C. Meyer, Matrix analysis and applied linear algebra, SIAM, 2000.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).