Andrew Larkin - Quenched decay of correlations for one-dimensional random Lorenz maps

This talk is based on my preprint Quenched decay of correlations for one-dimensional random Lorenz maps.

Given a dynamical system, we are often interested in whether we have a statistical property known 'decay of correlations' and at what rate the decay occurs, i.e. if we continue to compose the map on itself, then the system mixes and at a particular rate. Furthermore, given a random dynamical system, where we have a family of maps and take compositions of those maps, we are interested in knowing the statistical properties of the random dynamical system in a measure-theoretic sense, i.e. for almost every composition of maps from the family, we have decay of correlations. To prove this, one can use a tool known as a Tower.

The main focus of this talk is to explain what Towers are and how they can be used to obtain decay of correlations results for both deterministic and random dynamical systems. We start with a simple deterministic case and show how we construct a Tower and how it gives us decay of correlations. We then build up to the random case, which is more involved than the deterministic case, and show how it allows us to prove the main result of the preprint.