Lyapunov exponents for transfer operator cocycles of random one-dimensional maps

Non-autonomous or random dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. In the last fifteen years, transfer operators have been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of random dynamical systems, through the so-called Lyapunov--Oseledets spectrum. While the scope of this framework is broad, in practice it is challenging to identify and even approximate this spectrum. In this talk, we present examples of one-dimensional map cocycles where we can describe the Lyapunov--Oseledets spectrum and understand some of its stability properties under perturbation.