Invariant Measures for Uncountable Random Interval Homeomorphisms

• Janusz Morawiec ^[1] ; Tomasz Szarek ^[2]
  1. [1] University of Silesia

     University of Silesia

     Katowice, Polonia

     
  2. [2] University of Technology

     University of Technology

     Rusia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A necessary and sufficient condition for the iterated function system { f (·, ω)| ω ∈ } with probability P to have exactly one invariant measure μ∗ with μ∗((0, 1)) = 1 is given. The main novelty lies in the fact that we only require the transformations f (·, ω) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.

• Referencias bibliográficas
  • 1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd ed. Springer, Berlin (2006). A Hitchhiker’s Guide
  • 2. Alsedá, L., Misiurewicz, M.: Random interval homeomorphisms. Publ. Math. 58, 15–36 (2014)
  • 3. Barnsley, M.F., Demko, S.G., Elton, J.H., Geronimo, J.S.: Invariant measures for Markov processes arising from iterated function systems...
  • 4. Baron, K., Jarczyk, W.: Random-valued functions and iterative functional equations. Aequationes Math. 67(1–2), 140–153 (2004)
  • 5. Baron, K., Kuczma, M.: Iteration of random-valued functions on the unit interval. Colloq. Math. 37(2), 263–269 (1977)
  • 6. Benoist, Y., Quint, J.F.: Random Walks on Reductive Groups. Springer, Berlin (2016)
  • 7. Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd ed. Wiley,...
  • 8. Czudek, K., Szarek, T.: Ergodicity and central limit theorem for random interval homeomorphisms. Isr. J. Math. 239, 75–98 (2020)
  • 9. Da Prato, G.: An Introduction to Infinite-dimensional Analysis. Universitext. Springer, Berlin (2006). Revised and extended from the 2001...
  • 10. Diamond, P.: A stochastic functional equation. Aequationes Math. 15(2–3), 225–233 (1977)
  • 11. Doob, J.L.: Measure Theory. Graduate Texts in Mathematics, Vol. 143. Springer, New York (1994)
  • 12. Dudley, R.M.: Real Analysis and Probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge...
  • 13. Furstenberg, H.: Boundary Theory and Stochastic Processes on Homogenous Spaces. In: Proceedings of Symposia in Pure Mathematics, vol....
  • 14. Gharaei, M., Homburg, A.J.: Random interval diffeomorphisms. Discrete Continu. Dyn. Syst. 10, 241–272 (2017)
  • 15. Kapica, R.: Sequences of iterates of random-valued vector functions and continuous solutions of a linear functional equation of infinite...
  • 16. Kochanek, T., Morawiec, J.: Probability distribution solutions of a general linear equation of infinite order. Ann. Polon. Math. 95(2),...
  • 17. Malicet, D.: Random walks on homeo (s1). Comm. Math. Phys. 356(3), 1083–1116 (2017)
  • 18. Matkowski, J.: Remark on BV-solutions of a functional equation connected with invariant measures. Aequationes Math. 29(2–3), 210–213 (1985)
  • 19. Mihailescu, E., Urba ´nski, M.: Random countable iterated function systems with overlaps and applications. Adv. Math. 298, 726–758 (2016)
  • 20. Mihailescu, E., Urba ´nski, M.: Geometry of measures in random systems with complete connections. J. Geom. Anal 32(5), 162, 18 (2022)
  • 21. Morawiec, J., Reich, L.: The set of probability distribution solutions of a linear functional equation. Ann. Polon. Math. 93(3), 253–261...
  • 22. Morawiec, J., Zürcher, T.: A new take on random interval homeomorphisms. Fund. Math. 257(1), 1–17 (2022)
  • 23. Ott, W., Tomforde, M., Willis, P.N.: One-Sided Shift Spaces Over Infinite Alphabets. New York Journal of Mathematics, vol. 5. NYJM Monographs....
  • 24. Szarek, T., Zdunik, A.: Attractors and invariant measures for random interval homeomorphisms. Manuscript