In my talk I'd like to present recent results on the behavior of ergodic properties of random maps on an interval and the real line. We give some sufficient conditions (close to optimal) for the existence of a unique invariant ergodic measure. We provide also examples with many invariant ergodic measures. We will present also a general method that allows one to define random systems such that uniqueness fails.
The results have been obtained in collaboration with Sara Brofferio (Paris), Hanna Oppelmayer (Innsbruck): Unique ergodicity for random noninvertible maps on an interval, Annales l'Institut Fourier (2025).