Higher-order correlations for group actions

Abstract

This survey paper discusses behaviour of higher-order correlations for one-parameter dynamical systems and more generally for dynamical systems arising from group actions. In particular, we present a self-contained proof of quantitative bounds for higher-order correlations of actions of simple Lie groups. We also outline several applications of our analysis of correlations that include asymptotic formulas for counting lattice points, the existence of approximate configurations in lattice subgroups, and validity of the Central Limit Theorem for multi-parameter group actions.