turkmath.org

We consider probability-preserving dynamical systems from countable abelian group actions.  Such a system is said to be a Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor (these factors arise in the study of multiple recurrence and play a foundational role in related areas in additive combinatorics and number theory).  We provide a classification of Conze-Lesigne systems in terms of algebraic data.  More precisely, we show that an arbitrary Conze-Lesigne system is an inverse limit of translational systems arising from locally compact nilpotent groups of nilpotency class 2 quotient by a lattice.  Results of this type were previously known when the acting group is finitely generated or a direct sum of cyclic groups.  The talk aims at introducing the field.  If time permits, we will present an application to additive combinatorics.  The talk is based on recent joint works with Or Shalom and Terence Tao.