Randomness, dynamics and spectrum – ADYCT

The last fifteen years have witnessed several significant progresses on the analytical understanding of chaotic dynamical systems, and in parallel on the study of models of random waves. Even if they took place in parallel, these new developments share many similarities in their objectives and methods: asymptotic distribution of eigenvalues, existence of spectral gaps, decay of correlations, etc. Both these topics play a fundamental role in the study of quantized chaotic sytems, a domain colloquially referred to as quantum chaos. The aim of this proposal is to pursue the development of these topics, and develop the growing interactions between them. This will be achieved by bringing together young promising researchers and more experienced ones, and by focussing on three related tasks:

- Task 1. Random waves, hyperbolic dynamics and quantum chaos. In this first task, we will deepen our understanding of the random wave model, in particular with regard to applications to quantum chaos. We will also analyze the ergodic properties of classical chaotic systems, through the scope of their spectra of Ruelle resonances, mimicking the analysis of quantum sytems.

- Task 2. Weakly chaotic systems, cohomological equations and differential topology. In this second task, we will extend the study of Ruelle spectra to models with weaker chaotic properties, such as Teichmüller flows or Axiom A flows. We will also elaborate on the recent developments about the topological content of these Ruelle spectra.

- Task 3. Spectra of random operators. In this last task, we will group operators within probabilistic ensembles, thereby defining classes of random operators, and study their spectral properties in this probabilistic viewpoint. In particular, we will show how this randomness can help some questions raised in the first two tasks, e.g. existence of spectral gaps, asymptotic distribution of eigenvalues, properties of eigenfunctions, etc.