course: Minicourse 1 A semi-classical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics
speaker: Frédéric Faure (Université de Grenoble Alpes)abstract: Uniformly hyperbolic dynamics (Anosov, Axiom A) have "sensitivity to initial conditions" and manifest "determinist chaotic behavior", e.g. mixing, statistical properties etc. In the 70', David Ruelle, Rufus Bowen and others have introduced a functional and spectral approach in order to study these dynamics which consists in describing the evolution not of individual trajectories but of functions and observing the convergence towards equilibrium in the sense of distribution. This approach has progressed and these last years, it has been shown by V. Baladi, C. Liverani, M. Tsujii, N. Roy, J. Sjöstrand, S. Dyatlov, C. Guillarmou, M. Zworski, and others that the generator of this evolution operator ("transfer operator") has a discrete spectrum, called "Ruelle-Pollicott resonances" which describes the effective convergence and fluctuations towards equilibrium and that this spectrum is determined from the periodic orbits (using the Atiyah-Bott trace formula). Due to hyperbolicity, the chaotic dynamics sends the information towards small scales (high Fourier modes) and technically it is convenient to use "semiclassical analysis" which permits to treat fast oscillating functions. More precisely it is appropriate to consider the dynamics lifted in the cotangent space T*M of the initial manifold M (this is an Hamiltonian flow). We observe that at fixed energy (frequency along the flow direction), this lifted dynamics has a compact non wandering set called the trapped set and that the lifted dynamics on T*M scatters on this trapped set. Then the existence and properties of the discrete Ruelle-Pollicott spectrum follows from the uncertainty principle (i.e. effective discreteness of T*M) and rejoin a more general theory of semiclassical analysis developed in the 80' by B. Helffer and J. Sjöstrand called "quantum scattering on phase space".
The minicourse will explain some aspects of this topic and will have three parts:
1) For general Anosov flows we will explain the existence of an intrinsic discrete spectrum of Ruelle Pollicott resonances in specific anisotropic Sobolev spaces.
2) For the special case of contact Anosov flow, e.g. geodesic flow, we will explain that the above spectrum is structured into vertical bands.
3) We discuss some consequences of the previous results: relation between the spectrum and periodic orbits and in case of geodesic flow, the relation with the Laplacian operator and quantum chaos. This will suggest many open questions.
timetable:
Mon 27 Jun, 9:30 - 10:30, Aula Dini
Mon 27 Jun, 11:00 - 12:00, Aula Dini
Tue 28 Jun, 14:00 - 15:00, Aula Dini
Wed 29 Jun, 10:15 - 11:15, Aula Dini
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