Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## The joint spectrum

Prof. Dr. Emmanuel Breuillard talk

Speaker invited by: Prof. Dr. Corinna Ulcigrai

**Date:** 03.12.18 **Time:** 15.30 - 16.30 **Room:** Y27H25

The notion of joint spectral radius of a set S of matrices was introduced by Rota and Strang in the 60's and encodes the maximum asymptotic rate of spatial growth of a product of elements from S. It is intimately related to the maximal growth of eigenvalues of products of elements from S by theorems of Berger-Wang and Bochi. In this talk I will present a multi-dimensional version of this notion, where one looks at the full vector of eigenvalues leading naturally to the notion of joint spectrum of S. This is a compact subset of the Weyl chamber that can sometimes be explicitly computed and has connections with the asymptotic shape of large balls in Cayley graphs of Lie groups, with Lyapunov exponents for stationary processes, large deviations for random matrix products and with ergodic optimization. Joint work with Cagri Sert.