Aims of the research project.

The last fifteen years have witnessed several significant progress in the analytical understanding of chaotic dynamical systems, and at the same time in 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 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 focusing on three related tasks:

• 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 spectrum of Ruelle resonances, mimicking the analysis of quantum systems. 

• Weakly chaotic systems, cohomological equations and differential topology. In this second task, we will extend the study of Ruelle spectrum 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 this Ruelle spectrum.

• 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 from a probabilistic viewpoint. In particular, we will show how this randomness can help to address some questions raised in the first two tasks, e.g. existence of spectral gaps, asymptotic distribution of eigenvalues, properties of eigenfunctions, etc.