Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics.
Résumé
Uniformly hyperbolic dynamics (Axiom A) have ``sensitivity to initial conditions'' and manifest ``deterministic chaotic behavior'', e.g. mixing, statistical properties etc. In the 1970, 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 and others that this evolution operator (``transfer operator'') has a discrete spectrum, called ``Ruelle-Pollicott resonances'' which describes the effective convergence and fluctuations towards equilibrium. 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 $T^{\ast}M$ of the initial manifold $M$ (this is an Hamiltonian flow). We observe that at fixed energy, this lifted dynamics has a relatively compact non-wandering set called the trapped set and that this lifted dynamics on $T^{\ast}M$ scatters on this trapped set. Then the existence and properties of the Ruelle-Pollicott spectrum enters in a more general theory of semiclassical analysis developed in the 1980 by B. Helffer and J. Sj\"ostrand called ``quantum scattering on phase space''.
We will present different models of hyperbolic dynamics and their Ruelle-Pollicott spectrum using this semi-classical approach, in particular the geodesic flow on (non necessary constant) negative curvature surface $\mathcal{M}$. In that case the flow is on $M = T_{1}^{{\ast}}\mathcal{M}$, the unit cotangent bundle of $\mathcal{M}$. Using the trace formula of Atiyah-Bott, the spectrum is related to the set of periodic orbits. We will also explain some recent results, that in the case of contact Anosov flow, the Ruelle-Pollicott spectrum of the generator has a structure in vertical bands. This band spectrum gives an asymptotic expansion for dynamical correlation functions. Physically the interpretation is the emergence of a quantum dynamics from the classical fluctuations. This makes a connection with the field of quantum chaos and suggests many open questions.