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
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• 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.