The famous Baxendale Theorem states that for a random dynamical system by diffeomorphisms of a compact manifold M^d, unless the system possesses a measure that is invariant under all the maps of the system (that is quite rare), there exists an ergodic stationary measure with strictly negative «volume» Lyapunov exponent 
\lambda_vol = \lambda_1+…+\lambda_d. 
My talk will be devoted to a recent joint result with V. P. H. Goverse, generalising this theorem to a non-invertible (and only piecewise-continuous) setting. Now, the upper bound for the volume Lyapunov exponent is logarithm of the average number of preimages of a point. In particular, once this number does not exceed 1 (``\mu-injectivity’’ by Brofferio, Oppelmeyer and Szarek), the volume Lyapunov exponent can again be claimed to be negative.