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

TL;DR

This work develops a geometrical microlocal framework for contact Anosov flows, showing that high-frequency transfer operators are well approximated by quantum-type evolutions on the trapped symplectic set Σ. By constructing an anisotropic Sobolev space and a bundle calculus over E_s, N_s, and Σ, the authors prove that the Ruelle spectrum forms vertical bands with Weyl-law density in isolated bands and exhibit ergodic concentration within those bands. A central achievement is the emergence of an effective quantum dynamics on vector bundles, realized through a wave-packet (Bargmann) transform and a quantization Op_Σ of the classical dynamics on Σ, together with horocycle-operator analogues that shift band indices. These results connect deterministic chaotic dynamics to a quantum-like propagation, offering precise spectral descriptions and trace-type formulas that extend semiclassical quantization to nonconstant curvature settings. The framework unifies microlocal analysis, geometric quantization, and dynamical zeta-type structures to illuminate long-time behavior in chaotic systems and provides tools for Weyl laws and spectral concentration in the Ruelle spectrum.

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.

Figures (10)

• Figure 1.1: The dynamics induced on $T^{*}M$ scatters on the trapped set $\Sigma\subset T^{*}M$ defined in (\ref{['eq:def_Sigma']}). $\Sigma$ is a line bundle over $M$, a symplectic sub-manifold of $T^{*}M$ and at every point $\rho=\omega\mathcal{A}\left(m\right)\in\Sigma$, where $\omega$ called frequency is the coordinate along the line, the symplectic-normal bundle $N\left(\rho\right)=\left(T_{\rho}\Sigma\right)^{\perp}$ (a symplectic linear subspace of $T_{\rho}T^{*}M$) splits into unstable/stable subspaces, $N\left(\rho\right)=N_{u}\left(\rho\right)\oplus N_{s}\left(\rho\right)$, see (\ref{['eq:decomp_K_K0_N']}). The main geometrical object considered in this paper is this fibration $N_{s}\rightarrow\Sigma\rightarrow M$. Beware that for a geodesic flow on $\left(\mathcal{N},g\right)$, this fibration sequence continues with $M=\left(T^{*}\mathcal{N}\right)_{1}\rightarrow\mathcal{N}$.
• Figure 1.2: The dots represent the intrinsic Ruelle discrete spectrum of the derivation $X_{F}$ in a Sobolev space $\mathcal{H}_{W}\left(M;F\right)$. From faure_tsujii_Ruelle_resonances_density_2016, the essential spectrum is in a (brown) vertical band that can be moved arbitrarily far on the left by changing the weight $W$, and reveals this intrinsic discrete spectrum. The right most eigenvalues in the first band $B_{0}$ dominate the emerging behavior of $e^{tX_{F}}$ for $t\gg1$. On figure (b) for the special case of the bundle $F=\left|\mathrm{det}E_{s}\right|^{1/2}$, the first band coincides with the imaginary axis.
• Figure 1.3: Band spectrum of $e^{tX_{F}}$ in $\mathcal{H}_{W}\left(M;F\right)$ for $t>0$, from Theorem \ref{['thm:Discrete-band_spectrum-1']}. For $k\in\mathbb{N}$, and $\gamma_{k}^{\pm}$ defined in (\ref{['eq:def_gamma_-2']}), the operator $e^{tX_{F}}$ has discrete spectrum in the annulus $e^{t\gamma_{k+1}^{+}}<\left|z\right|<e^{t\gamma_{k}^{-}}$ (if $\gamma_{k+1}^{+}<\gamma_{k}^{-}$) called “ spectral gap” . It has also discrete spectrum on $\left|z\right|>e^{t\gamma_{0}^{+}}$ and possibly essential spectrum elsewhere (the colored annuli). This picture is somehow the exponential of Figure \ref{['fig:Band-structure-of']}.
• Figure 3.1: Illustration of Lemma \ref{['lem:KN']} about the vector bundle $\widetilde{\mathrm{Ker}\mathcal{A}}\subset T_{\Sigma}T^{*}M\overset{\pi}{\rightarrow}M$. We have the linear map $\widetilde{\mathrm{Ker}\mathcal{A}}\overset{d\pi}{\rightarrow}\mathrm{Ker}\mathcal{A}\subset TM$ and $\widetilde{\mathrm{Ker}\mathcal{A}}$ is a sum of two symplectic subspaces $K\oplus N$ and two Lagrangian subspaces $H\oplus V$. This structure is preserved by the dynamics.
• Figure 4.1: Anosov flow $\phi^{t}$ (in solid line) generated by a vector field $X$ on a compact manifold $M$.

Theorems & Definitions (236)

• Remark 1
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Emergence of quantum dynamics
• Remark 1.6
• Theorem 1.7: Band structure of the Ruelle spectrum
• Theorem 1.8: Ring spectrum
• Remark 1.9
• Remark 1.10
• Theorem 1.11: Weyl law for isolated bands