Invariant Measures for Uncountable Random Interval Homeomorphisms

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