Non-Uniform Synchronization in Random Dynamical Systems

The goal of the project is to investigate the transient behavior of discrete-time random dynamical systems before synchronization in the negative Lyapunov exponent regime.

The setup is as follows: we fix a family of contracting and expanding linear maps and vary the probabilities such that the Lyapunov exponent of the iterated function system remains negative. The problem concerns the phase transition to chaos: as the Lyapunov exponent approaches zero through varying probabilities, we study the distribution of "escapes from synchronization" - a new random variable derived from the two-point motion of the system. This variable counts the finite number of exits from a small ball around the diagonal before the system synchronizes near the diagonal.

In one-dimensional systems, we prove an inverse relationship between the expected number of escapes and the Lyapunov exponent. Interestingly, the statistics depend only on the probabilities via the Lyapunov exponent. Our aim is to generalize this relationship to higher dimensions, particularly to the case of compositions of linear transformations in Euclidean spaces. In the future, we also plan to consider the case of non-linear maps.