**Title:** A discrete view of correlations: Ruelle resonances and their stochastic stability

**Speaker:** Maciej Zworski

**Speaker Info:** Berkeley

**Brief Description:**

**Special Note**:

**Abstract:**

The poles and zeros of zeta functions from Lecture 2 appear in expansions of correlations, $ \rho_{f,g} ( t ) $, or as poles of their power spectra, $ \widehat\rho_{f , g } ( t) $: \[ \rho_{f,g}( t ) := \int_M f ( \varphi_{-t} ( x ) ) g ( x ) dx , \ \ \ \ \widehat \rho_{f,g} ( \lambda ) := \int_0^\infty \rho_{f,g}(t ) e^{ i\lambda t } d t , \] where $ \varphi_t : M \to M $ is an Anosov flow and $ f $ and $ g $ are test functions, say $ f , g \in C^\infty ( X ) $. These poles are called {\em Ruelle} or {\em Pollicott--Ruelle} resonances and they are ``discrete signatures of chaos''.For instance, in this language {\em exponential decay of correlations} is equivalent to having an {\em essential spectral gap}: a strip with finitely many resonances. Existence of this gap (in successive generality and precision) was shown for contact Anosov flows by Dolgopyat in 1998, Liverani in 2004, Tsujii in 2012 and, in a way also applicable to quantum systems, by Nonnenmacher and the speaker in 2015. On the other hand in joint work with Jin 2016, we showed that for any Anosov flow the essential spectral gap is finite.

To have physical meaning Pollicott--Ruelle resonances have to be stable under stochastic perturbations, that is, after adding {\em Brownian motion} to the flow $ \varphi_t $. That can be done in different ways and for homogeneous Brownian motions and any Anosov flows Dyatlov and the speaker did it in 2015. A more subtle result was obtained by Drouot in 2016 for {\em kinetic Brownian motion} on Riemannian manifolds. In a fascinating recent development Dang--Riviere applied microlocal methods to the study of Morse--Smale flows. In that case stochastic stability is related to the spectral properties of the Witten Laplacian and to the 2011 work of Frankel--Losev--Nekrasov on instantons.

Explaining some of this will be the purpose of the last lecture.

Copyright © 1997-2024 Department of Mathematics, Northwestern University.