Mathematical Colloquium: Tomas Persson (Lund University)

Title

Linear response

Abstract

Consider a family of chaotic dynamical systems that depends on a parameter t. In case time is discrete, a dynamical system can be described by iterating a function T:M→M, which depends on the parameter t. The orbit of an initial state x is the sequence x, T(x), T(T(x)), … and for chaotic systems, small perturbations of the inital point x quickly leads to completely different orbits. However, for "most" initial points x, the orbits are distributed over M in the same way, although the orbits are very different.

The typical distribution of an orbit in M can be described by a probability measure on M. "Linear response" means that this measure in some sense depends in a differential way on the parameter t. The mathematical theory of linear reponse goes back to David Ruelle. It came originally from statistical physics, but has also found applications in climate modeling.

To understand linear response, we will need a bit of ergodic theory. I will start by explaining the basics of ergodic theory and then give an overview of the theory of linear response, and mention both old and new results. Towards the end I hope to say a few words about ongoing research.