Random dynamical systems in real and complex variables

In this thesis, we study the long-term behavior of different types of random dynamical systems, in real and complex variables. We consider systems such as random walks and iterated function systems (IFS). We start with a chapter studying a non-linear dynamical system in a setting with real variables. Non-linear dynamics can get complicated, especially in higher dimensions. To still describe which phenomena can occur, we use in the last two chapters the rigidity of holomorphic dynamics in one complex variable. Each chapter is dedicated to a specific system, so they can be read independently from each other.