# DYNRIGDIOPHGEOM Sintesi della relazione

Project ID:
239606

Finanziato nell'ambito di:
FP7-IDEAS-ERC

Paese:
United Kingdom

## Final Report Summary - DYNRIGDIOPHGEOM (Dynamics of Large Group Actions, Rigidity, and Diophantine Geometry)

This project was devoted to study asymptotic properties of chaotic multi-parameter group actions and involved three important research directions: distribution of orbits, behaviour of order correlations, and rigidity of actions. The proposed research was motivated in part by some open problems in Number Theory. Distribution of complicated orbits can encoded by suitable averaging operators. We have established an asymptotic formula for a general class of the averaging operators which is not only of fundamental importance in the Theory of Dynamical Systems, but also found important applications to Number Theory. In particular, we have used averaging operators to prove asymptotic counting formula for the number of lattice points in algebraic groups. Furthermore, applying these techniques, we have developed a sieving argument for counting lattice points with prescribed arithmetic properties. Another significant application of averaging operators to Number Theory, which we explored in this project, involves the problem of Diophantine approximation on algebraic varieties. We have established bounds for exponents of Diophantine approximation on homogeneous algebraic varieties which are sharp in a number of cases. We have also explored properties of higher order correlations for actions on nilmanifolds by automorphisms and proved explicit quantitative estimates for convergence of these correlations. In the direction of rigidity of group actions, we considered the problem of classification of measurable maps which are equivariant under actions of discrete groups and proved under quite general conditions every such measurable map is smooth and, in fact, equivalent under the action of a larger connected group. We also explored rigidity under perturbations of actions of discrete groups on boundaries.