A new take on random interval homeomorphisms

The purpose of this paper is to extend a result on the existence and uniqueness of an invariant measure of random interval homeomorphisms obtained recently by T. Szarek and A. Zdunik. We consider IFS's generated by orientation preserving homeomorphisms of [0, 1]. Each such IFS has two trivial invariant measures supported at the points 0 and 1. We give a necessary and sufficient condition for the existence and uniqueness of a third invariant measure mu* with mu*((0, 1)) = 1. Our condition does not require the existence of the Lyapunov exponent at 0, nor the existence of the Lyapunov exponent at 1. Moreover, in contrast to the result of T. Szarek and A. Zdunik, it can be applied also to many "critical" IFS's for which both the Lyapunov exponents at 0 and at 1 vanish. The main idea of this paper is based on an elementary application of functional equations.