We consider a stochastic dynamical system on Euclidean space, where a point evolves by successive application of random transformations. Starting from an initial point x, we define the process by repeatedly applying independent, identically distributed random functions. A central question is whether changes in the starting point affect the long-term behavior of the process. In particular, we ask whether the distance between two trajectories—starting from distinct points—tends to zero as time evolves. We focus on systems that are not globally stable but show “local stability”: the trajectories become close to each other when we look at the process through a bounded window. While this behaviour has been observed in one-dimensional systems under certain critical conditions—neither strictly contracting nor expanding—it is much less understood in higher dimensions.
In this talk, we present recent results on local stability for multidimensional random affine recursions. These are defined by applying at each step a random linear transformation followed by a random shift. This process has been widely studied for its numerous applications and for its interest in the study of probabilities on algebraic structures. In fact its properties are strictly related to the behaviour of the product of random matrices. We focus on the critical regime, where the average rate of expansion or contraction (the Lyapunov exponent) is zero.
