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

TL;DR

This paper develops a micro-local analysis of contact Anosov flows, revealing the band structure of the Ruelle spectrum and its asymptotic distribution, with implications for spectral theory and dynamical systems.

Contribution

It introduces a geometrical micro-local framework for contact Anosov flows and characterizes the spectral band structure of the Ruelle spectrum in the high frequency limit.

Findings

Ruelle spectrum forms vertical bands in the complex plane.

Most spectrum in the right-most band concentrates along a line parallel to the imaginary axis.

The density of eigenvalues follows a Weyl law as the imaginary part increases.

Abstract

We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesic flow on negatively curved manifold. This micro-local analysis is based on wave-packet transform discussed in arXiv:1706.09307. 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 gives 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 concentrate along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tend to infinity. Some of these results were announced in arXiv:1301.5525.