Resumen de Invariant Measures for Uncountable Random Interval Homeomorphisms

Janusz Morawiec, Tomasz Szarek

• 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.