Microlocal Methods in Dynamical Systems (University of Zurich)

Abstract

The goal of this course is to introduce the powerful tools of Microlocal Analysis, and to present some striking applications in Chaotic Dynamical Systems.

Objective

Understanding the powerful concepts of Microlocal Analysis useful in Analysis and PDEs. Learning about the state-of-the-art applications to Dynamical Systems: anisotropic Sobolev spaces, and their applications to Ergodicity and Mixing properties, Zeta Functions.

Content

-Microlocal Analysis studies singularities of distributions in phase space, by describing the behaviour of a singularity in both position and direction. It is a part of the field of partial differential equations (PDE), created by Hoermander, Kohn, Nirenberg, and others in 1960s and 1970s, and is used to study questions such as solvability, regularity, and propagation of singularities of solutions of PDEs. To name a few classical applications: asymptotics of eigenvalues for elliptic operators (Weyl law), trace formulas, and inverse problems. -There have been recent exciting applications of Microlocal Analysis to Dynamical Systems and Geometry. These range from Dynamical Zeta Functions, Resonances and decay of correlations, to injectivity properties of X-ray (geodesic) transforms, and applications to Rigidity questions in Geometry. -The goal of this course is to introduce the powerful tools of Microlocal Analysis, and to present some striking applications in Chaotic Dynamical Systems. Here are the details (subject to changes): 1. Distributions and Fourier Transform (recap). Symbol classes and Oscillatory Integrals. Fourier Integral Operators. Stationary Phase Lemma. (3 lectures) 2. Pseudodifferential Operators (PDO). Compositions, changes of coordinates, calculus of PDOs. PDOs on manifolds. (2 lectures) 3. Elliptic regularity. L^2-continuity. Sobolev spaces and PDOs. (2 lectures) 4. Wavefront set. Products, pullbacks of distributions. (1 lecture) 5. Applications: 1) construction of anisotropic Sobolev spaces for chaotic dynamics, existence of Pollicott-Ruelle resonances. 2) Ergodicity and Mixing. 3) Possible applications: Ruelle Zeta Function, exponential Mixing for contact Anosov flows, Frame Flows and Parry’s representation. (6 lectures)

Resources