Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## e-Seminar: Noncommuting random products and metric functionals

Talk by Prof. Dr. Anders Karlsson

**Date:** 11.05.20 **Time:** 13.45 - 14.45 **Room:** Y27H28

The study of the composition of random maps that do not commute started in the 1950s with the need to solve linear difference equations with random coefficients, giving rise to products of random matrices (today appearing in the study of quasi-periodic Schrödinger operators). It also arises in differentiable dynamics, with the derivative cocycle. In both these settings a basic fundamental theorem is the multiplicative ergodic theorem of Oseledets asserting the existence of Lyapunov exponents. But there are several non-linear settings of interests: random walks on groups, surface homeomorphisms, deep learning, etc. In joint works with Ledrappier and with Gouëzel we established a general extension of the multiplicative ergodic theorem, that in particular provided an answer to a question raised in a seminal paper by Furstenberg from 1963. It is given in terms of metric functionals (which include horofunctions) and suggests a (non-linear) metric functional analysis / spectral theory.