Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum

We develop a geometric micro-local analysis of contact Anosov flow, such as the geodesic flow on negatively curved manifold. This micro-local analysis is based on the wavepacket transform discussed by Faure and Tsujii [Ann. H. Lebesgue 6 (2023), pp. 331–426]. The main result is that the transfer operator is well approximated (in the high frequency limit) by the quantization of the Hamiltonian flow naturally defined from the contact Anosov flow and extended to some vector bundle over the symplectization set. This has a few important consequences: the discrete eigenvalues of the generator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it concentrates along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tends to infinity. Some of these results were announced by Faure and Tsujii [C. R. Math. Acad. Sci. Paris 351 (2013), pp. 385–391].